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# Electronics Primer

20.309: Biological Instrumentation and Measurement

## Circuit models

The physical quantities that characterize many types of systems are defined at all points in space and time. For example, Maxwell's equations describe the relationship between the electric field, magnetic field, and charge at every point in space and time. The equations govern the operation of electric circuits. Cmputing the field and charge at every point is a very thorough approach to analyzing an electrical system. However, such a detailed model includes far too much information to be practical.

Analysis of many systems can be dramatically simplified by using circuit models. Circuit models characterize a system only at a small number of well-defined points by collapsing regions of space into lumped elements. A circuit model is simplified representation of a system that consists of a network of lumped elements connected together by wires. Regions of space that do not significantly impact system behavior are ignored.

Resistor symbol with terminal parameters defined.

A lumped element is an idealized abstraction of a physical component, such as a resistor. Elements are bracketed by terminals. Every element obeys a constitutive equation that describes the relationship between the state variables at its terminals. In elements with two terminals, one is typically designated (+) and the other (-). Each type of element has an associated graphic symbol. Symbols can be arranged in schematic diagrams to depict a system of interconnected elements. The symbol for a resistor is shown at right.

The state variables for an electric circuit are voltage (v) and current (i). Current measures the flow rate of electrons moving through the element. Voltage quantifies the electrical potential difference between two points. Higher voltages attract electrons more intensely and thus result in larger current flows. Every element has a single voltage across it and current flow through it.[1] The constitutive equation for an ideal resistor is given by Ohm's law, v=iR.

## Review of electrical units and equations

• Electric charge is measured in Coulombs, usually abbreviated with a capital C. Electrons have a charge of about 1.610×10−19 C. (One mole of electrons has a charge of about 96500 Coulombs.)
• Electric potential difference has the units of energy per charge and is measured in Volts. 1 Volt = 1 Joule/Coulomb = 1 kg m2 / sec 2/Coulomb. The variable v is most often used for voltage.
• Current flow has the units of charge per time and is measured in Amps. 1 Amp = 1 Coulomb / sec. [2] The variable i is normally used for current.
• Resistance has units of Ohms. 1 Ohm = 1 Volt divided by 1 Amp. Resistance is usually denoted by R
• Electric power is equal to voltage times current, P = v i. Power has units of energy per time called Watts. 1 Watt = 1 Volt x 1 Amp = 1 Joule / second.

### Networks of lumped elements

Table 1 lists the name, symbol, constitutive relation and its graphical representation for four ideal elements: a voltage source, a current source, a resistor, and a conductor. Each constitutive equation and associated plot relate voltage (v) across an element and current (i) through that element.

 Table 1: Constitutive relations for ideal elements.

An ideal voltage source maintains a constant potential difference across its terminals regardless of the amount of current flowing through it. Thus, its voltage versus current graph is a horizontal line. The current flowing through an ideal current source is constant no matter what potential difference is across its terminals. The ideal resistor obeys Ohm’s law, v = iR — a sloped line on the voltage versus current graph. Ideal conductors have zero resistance, meaning that the voltage at all points on the conductor is the same for any value of current flow.

To form a network of elements, terminals are connected together by ideal wires. Elements interact only through their terminals. Terminals may be shown as filled or open circles to indicate whether or not that terminal connects to another electrical element. (Open circles represent no connection.) Pairs of terminals are sometimes called ports, whether for a single element or across multiple elements. A port approach is particularly useful for abstracting away (black-boxing) multiple circuit elements and treating them as an equivalent element with defined inputs and/or outputs. We'll return to this idea in sections 5.4 and 5.6.

A voltage v and current i can be defined at any terminal. For terminals A and B in Figure 1, the element voltage v = vAB = vAvB and the element current i = iA = iB. Here, vA and vB are the absolute potentials (defined relative to a reference), and v is the potential difference between the two terminals.

Two limiting cases of the resistor element are useful for solving circuit problems: the open circuit and the short circuit (Fig. 2). Infinite resistance means that no current passes through that element; this is also called an open circuit across said element and is equivalent to an unconnected port. Zero resistance means no potential difference appears across the element; this is also called a short circuit across that element and is equivalent to an ideal wire.

Figure 2: Open and short circuits. In the open circuit (left), the maximum potential difference exists across the resistor, and VAB = V. In the short circuit (right), the potential is the same everywhere because a wire connects the two resistor terminals, and VAB = 0.

### Illustrative circuit analogies

We are now in a position to appreciate some analogies to electronic circuits. We’ll only describe in detail the most physically intuitive one, while the rest are listed in Table 2.

Table 2: Circuit variables in other engineering disciplines.

In a hydraulic system (Fig. 3), pressure differences between two points are analogous to voltages in electronic circuits. A water reservoir is like a voltage source, and water flow is analogous to current flow. The higher the pressure differential (greater amount/height of water in the reservoir), the faster the volumetric flow, all else being equal. Resistance can be thought of as the narrowness of the conduit connecting two points. The narrower the pipe, the slower the flow, all else being equal. Or it can be thought of as a pipe packed with beads. The smaller the beads, or the greater in number, the slower the flow, all else being equal.

Figure 3: Fluid dynamics analogy for circuits. At left, the (pipe) resistance is high, slowing water flow from the reservoir, but the voltage equivalent is also high, quickening water flow compared to a less full reservoir. At right, the parameters are just the opposite.

## Real electronics

The idealized resistor model is an excellent approximation of a real resistor's behavior under most circumstances.

## Connecting circuit elements

Multiple elements may be joined at their terminals to build a circuit (Fig. 4). The joined terminals are now called nodes. Any path between two (independent -- see below) nodes is called a branch. Any path from one node back around to itself is called a loop. We’ll revisit these terms when we solve circuit problems in section 5.

Figure 4: Circuit illustrating definitions for node, branch, and loop.

Note that two (or more) apparent nodes that are connected by an ideal conductor are considered equivalent, because the conductor by definition cannot maintain a potential difference. Thus, A and the (A)'s above represent only one independent node, not three, as do B and the (B)'s. The two Steve's are of course independent nodes :)

Now is a good time to emphasize the difference between an element variable and a node variable. The element voltage vAB is the difference between the two node voltages, vA and vB. In other words, an element voltage equals the potential difference across two terminals, while a node voltage equals the absolute potential at a terminal. Of course, this "absolute" potential must still be with respect to a common reference point.

### Series and parallel

When two elements are joined end to end, they are said to be in series. In this case, the (-) terminal of one element is joined to the (+) terminal of the second element. If instead the (+) terminals of the two elements are connected, as are their (-) terminals, then the elements are said to be in parallel.

Figure 5: Resistors in series and parallel (with respect to one voltage source).

Resistors in series are additive: R = R1 + R2.

Resistors in parallel are inversely additive: $\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$, and thus $R = \frac{R_1 R_2}{R_1 + R_2}.$

These relationships are consistent with resistance being proportional to length and inversely proportional to area for a given material. The current that flows through each resistor in series must be identical, while for resistors in parallel, the voltage across each one must be identical. We will learn why when we discuss Kirchoff’s laws in section 5.1.

Similarly, voltage sources connected in series are additive, with current unaffected. Voltage sources linked in parallel maintain the potential difference of a single voltage source (so all sources must be identical or things get tricky ), with increased current flow.[3]

Series and parallel relationships can be used to reduce complex circuits to more simple ones as an intermediate step in problem-solving.

### Ground

Note that a voltage, or potential difference, is always a relative description. For example, for a voltage source the nominal (supposed) voltage -- say, 12 V -- is the potential difference between the (+) and (-) terminals. It is usually really convenient when solving circuit problems to define the (-) terminal as zero. (The relative nature of voltage allows this arbitrary choice.) This potential level is called ground. Ground can be considered a reservoir of electrons. By definition, any terminal connected to ground by a wire must also be at 0V. Ground is also the usual reference point for defining terminal/node voltages.

Figure 6: Multi-component circuit with all (dependent) grounded nodes marked.

However, there is nothing stopping us from defining the (+) and (-) terminals as +6V and -6V, and this may be convenient in certain problems.

### Understanding equivalent diagram topologies and notations, part 1

A common group of elements comprises a voltage source and two resistors in series. For reasons we shall see shortly, this grouping or "primitive" is called a voltage divider. Two common ways to depict this circuit are shown below.

Figure 7B: Voltage divider circuit, separate grounds.
Figure 7A: Voltage divider circuit, complete connections.

We can see that the voltage at node C is trivially 0, and that at node A is the maximum source voltage, because these are connected to the voltage source by ideal wires. The voltage at node B depends on the relative values of the two resistors.

As an example, let's take V = 12 V, R1 = 200 Ω, and R2 = 600 Ω. Each resistor is described by v = iR, where i must be uniform in this purely series-connected circuit. Ohm's law also applies to the circuit as a whole when using the overall series equivalent resistance Req, namely the sum of the two resistances. Thus, v = 12V = iReq = i(R1 + R2) = i(200 + 600)Ω, which can be solved for i = 15 mA.

The first resistor is described by vAB = iR1 = 15mA * 200Ω. Thus, it has an element voltage of vAB = 3V, and the node joining the two resistors has node voltage vB = 12 − 3 = 9V. Indeed, solving the second resistor equation vBC = iR2 = 15mA * 600Ω yields 9 V (with vC= 0 , of course).

Clearly, a voltage divider "divides" the voltage drop across resistors in series in proportion to their resistances. In sections 5.2-5.3 we will see a more universally applicable way to solve this problem, when we cannot rely on our intuition as much. Then we will imagine connecting measurement devices to the circuit and also further discuss topology.

Some other examples of topogically equivalent circuits are shown below (Fig 8).

Figure 8B: Topological equivalence, 3 resistors.
Figure 8A: Topological equivalence, 2 resistors.

## Putting it all together: solving circuit problems

### Solving problems: fundamentals/theory

Two key relationships provide the foundation for circuit analysis: Kirchoff’s circuit law and Kirchoff’s voltage law, hereafter KCL and KVL. KCL is based on charge conservation: at any node, the sum of the current magnitudes flowing in must equal the sum of the current magnitudes flowing out. When solving problems in which current directions are trivially known, this form of the law works well.

And now we see why current must be equal for elements connected in series: at the shared node, current coming out from the first element must have the same magnitude as current coming into the second element. No other branches exist at that node.

Returning to the issue of signs, current always flows from the (+) to the (-) terminal for an element, that is, from higher to lower voltage (except in a voltage source). However, the terminal definitions may not always be obvious/known in advance of solving the circuit problem. To keep signs straight, it will be convenient to have a systematic relationship between i and v signs. This relationship is described in the problem-solving section just below.

KVL is based on conservation of energy: the sum of the element voltages (potential difference across each element) around a loop is zero. Equivalently, the potential difference between the starting and ending node in a loop (that is, the same node) must be zero.

Take, for example, the three node voltage divider from section 4.3. Applying KVL directly in a clockwise direction, vAB + vBC + vCA = 3 + 9 − 12 = 0. KVL is even more straightforwardly applied using node voltages. Again going clockwise, clearly node voltages vAvB + vBvCvA + vC = 0, without even plugging in numbers.

Now we also see why voltage must be equal for elements connected in parallel: the two shared nodes form a loop, and hence the potential differences across the two elements must be identical.

### Solving problems: strategy

One rigorous way to solve circuit problems is to write the constitutive relation for each element (B equations for B branches), every independent KCL statement (N-1 equations for N nodes), and every independent KVL statement (B-1+N equations for B branches). Then algbraic substitutions are made until every element voltage and current is known. Often it is not obvious which substitutions will most quickly yield the quantities that are actually of interest in that problem.

A faster universal approach for solving circuit problems indirectly incorporates KVL and requires fewer KCL statements. Rarely can there be confusion about which choice of equations will quickly yield meaningful results. (For very complex circuits, however, linear rather than simple algebraic approaches may be required.)

Recall that while an element voltage is the potential difference across said element, a node voltage is the potential difference between that node and a common reference point – usually ground. (If these definitions doesn't sound familiar, see Figure 1 under Primary definitions and also review Connecting circuit elements.)

Figure 9: Voltage divider with current flow convention shown. The true flow of the current in this circuit is clock-wise. Hence, i2 would be negative because the assumed toward-node direction is opposite the true flow.

Working with node voltages rather than element voltages makes KVL implicit, as we saw in the voltage divider example above. The basic approach is then to write a single KCL equation per unknown node voltage, while immediately substituting in the element constitutive laws in terms of voltages and resistances.[4]

Now we make our alternative statement of KCL: the sum of all currents flowing into a node is zero, where current now includes both magnitude and sign. For i = v / R relationships, always subtract the close node voltage from the far node voltage — VfarVclose — to calculate the element voltage v in the numerator, and the sign of the calculated current should always come out right[5]. It will be positive if flowing in as assumed, and negative if actually flowing out. Using this sign convention, a current pointing toward the node is positive, and one pointing away fom the node is negative. However, a voltage source whose far node is the negative terminal must yield a positive current, because a source does work to push charge against the usual direction. [Last part maybe could be phrased more clearly]

### Understanding equivalent diagram topologies and notations, part 2

Let’s return to the two-resistor voltage divider described in section 4.3 above to practice our systematic approach for solving circuits, calling the single non-trivial node of interest B for consistency. We can apply KCL to calculate it:

$i_1 + i_2 = 0 = \frac{12-v_B}{200} + \frac{0-v_B}{600} \to \text{nominal}\ v_B = 9 \text{V}.$

Now let’s imagine connecting a voltmeter to measure the voltage at node B. To read the correct relative potential, the voltmeter must also be connected at ground. In other words, the second resistor and the voltmeter with associated resistance Rm are in parallel. Before attaching the voltmeter, the access port of interest can be shown as:

Figure 10: Voltage divider with access port for voltmeter highlighted.

After attaching the voltmeter, three equivalent circuit diagrams depicting the system are:

Figure 11: Voltage divider with voltmeter attached.

The voltmeter must have a high enough resistance that it does not draw current and substantially affect vB. If instead it has a comparable resistance, say 300 Ω, let’s calculate the effect on vB using our usual KCL approach:

$i_1 + i_2 + i_3 = 0 = \frac{12-v_B}{200} + \frac{0-v_B}{600} +\frac{0-v_B}{300} \to \text{nominal}\ v_B = 6 V.$

This value is unacceptably far off from our nominal 9 V. However, if Rm is increased an order of magnitude, to 3 kΩ, vBis a respectable 8.57 V. (Try it!)

Wouldn't it be nice to calculate vB for a given Rm without repeating the full calculation each time? The next section describes just such a method.

### Thevenin equivalent circuits

Topological equivalence between circuits implies that wires have been moved around or added in place of shorthand. There is also such a thing as functional equivalence between circuits. In fact, any complex circuit consisting only of voltage sources and resistors can be reduced to a circuit with only a single voltage source and resistor! This equivalent circuit cannot be empirically distinguished from the multi-component one.

Let’s return to the voltage divider example with a port across the second resistor. We want to know how the circuit as a whole responds at this access port when there is no load on it, i.e., no element in which energy can be dissipated. We calculate the equivalent circuit by three steps.

First we calculate the open circuit voltage (Voc) at that port, namely 9 V as above in section 5.3.

Next we calculate the short circuit current. In this hypothetical, the attached voltmeter has no internal resistance and behaves as a wire (Fig 12). In turn, vB is shorted to ground, and current magnitude is dependent only on R1, not both R1 and R2. Hence, the short circuit current Isc = 12V / 200Ω = 0.06A = 60mA.

Finally, the Thevenin resistance $R_{Th} = \frac{V_{oc}}{I_{sc}} = \frac{9 \text{V}}{0.06 \text{A}} = 150 \Omega$. The Thevenin equivalent circuit is shown in Figure 13.

Now how does this two-component circuit reduction help us? The voltmeter (or any element!) can be connected in series with RTh, and vB calculated immediately by the simple voltage divider relation, $\frac{V R_m}{R_{tot}}.$

An alternative and direct calculation of RTh is the total resistance at the port as viewed looking into the circuit (with voltage sources changed to wires): here, R1 and R2 in parallel.

Figure 12: Short circuit current for voltage divider.
Figure 13: Thevenin reduction of voltage divider. The Thevenin equivalent of the previously described voltage divider is shown (left) with an open port ready to be connected to a voltmeter with arbitrary resistance (right).

### Alternative problem-solving strategies

Circuit problems can be approached in many ways[6] and in the interests of time we will primarily treat the approach that (nearly) universally works, namely the node voltage/KCL method described above. Let’s briefly look at one example of another approach.

Three resistor circuit.

Consider the circuit shown at right. The parallel resistors R2 and R3 can be temporarily collapsed into an equivalent single resistor. Its resistance is given by

$R_{parallel} = \frac{R_2 R_3}{R_2 + R_3} = 200 \Omega.$

Then, the current through the purely series-connected circuit is simply

$i = \frac{V}{R_{tot}} = \frac{V}{R_1 + R_{parallel}}$

where Rtot = 400Ω and by the voltage divider relation vA is clearly 6 V. That is, half the voltage drop occurs across each 200 Ω resistor, the physically real one and the imaginary one (equivalent to two real ones).

We next expand the circuit to once more include the real, parallel branches. The potential difference across each branch must be the same vA, and thus the individual currents can be calculated from vA = ibranchxRbranchx.

For example, $i_2 =\frac{v_A}{R_2} = \frac{6}{300} = 20 \text{mA}$.

As expected, i2 and i3 sum to the current through the first resistor.

Yet another general circuit-solving approach is applying energy conservation equations.

### Limits of ideal elements: more realistic model of a battery

Recall the ideal i-v curves for our four elements, referring to Table 1 under Primary definitions .

Let’s model a battery as an ideal voltage source connected to no other elements except ideal wires (Fig. 14).

What is the current flowing through this circuit? The resistance of the wires is zero, implying that the current is infinite! Therefore this model, and the associated horizontal i-v curve, is not realistic.

Figure 14: Model for ideal battery.
Figure 15: More realistic battery model with access port.

Instead, let’s model the battery as having an internal resistance, RBATT (Fig. 15). To investigate the i-v curve, we’ll view the output port formed by nodes A and B.

When the port is not connected to anything, resistance is infinite and current is zero. There is no load on the battery, and thus the output voltage at node A must be the full VBATT. This useful quantity is our by now familiar open circuit voltage, Voc.

Next, let’s attach a voltmeter to the port with an associated resistance Rm.

Figure 16: More realistic battery model with voltmeter attached.

When we assume that Rm = 0, vA is shorted to ground. Thus, the current i is at its maximum, namely the short circuit current:

$I_{sc} = \frac{V_{BATT}}{R_{BATT}}$.

These two quantities define two points on our new, more realistic, i-v curve! Presumably a line connects them, but let’s prove it by solving this simple circuit. We’ll practice our systematic approach once again.

The only unknown node voltage is vA. The other nodes are either ground or VBATT by definition of a perfect conductor. Hence we want to solve KCL at VBATT, substituting in the relevant v = iR constitutive relations:

$i1 + i2 = 0 = \frac{9-v_A}{R_{BATT}} + \frac{0-v_A}{R_m} = 0$

and substituting for Rm using $I = \frac{v_A}{R_m}$

to simplify to vA = 9 − I * RBATT,

indeed the equation of a line.

Figure 17: More realistic battery model: i-v plot.

From the model diagram, we see that a real battery only sources a full 9V when no load is applied. As current increases, supplied voltage decreases linearly, with negative slope equalling internal resistance. Here is just one simple example of how we can learn about a system by perturbing it -- whether theoretically or experimentally. In this case we found the function relating voltage and current by imagining putting a load on the battery. Note that the equation we found is ultimately not dependent on the resistance of this added load Rm.

Finally, note that the symbol for a real battery is different from that of the ideal voltage source shown earlier. A typical battery symbol looks like Figure 18A; an arbitrary number of additional lines may be shown in some cases (since most batteries actually contain multiple electrochemical cells), as in Figure 18B.

Figure 18B: Multi-cell battery symbol from public domain.
Figure 18A: Single-cell battery symbol from public domain.

Thanks for reading! I'll appreciate any constructive feedback about the utility of this guide. Agi Stachowiak 21:11, 11 October 2012 (EDT)

## Circuit analogies

Circuit models are useful for solving problems in many engineering domains — electrical, mechanical, thermal, hydraulic, acoustic, and several others. Anything that is well described by a system of linear ordinary differential equations is amenable to circuit analysis. For simplicity, we’ll start with circuit elements whose behavior is not frequency dependent, namely resistors and sources, and build up to more complex systems involving capacitors and inductors in lecture/lab and perhaps someday a subsequent primer.

## References

1. For more about circuit assumptions and abstraction layers, see Agarwal 1.1-1.4: The power of abstraction, The lumped circuit abstraction, The lumped matter discipline, and Limitations of the lumped circuit abstraction.
2. By convention, current is positive in the direction of positive charge flow, which means the electrons are actually flowing in the negative direction.


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# Basic College Mathematics: An Applied Approach (Special

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However, most experimental studies of irregular reflection have used supersonic wind tunnels or shock tubes, and few or no data are available for Mach reflection phenomena in the plasma regime , e.g. Arithmetic a Text Workbook http://www.patricioginelsa.com/lib/arithmetic-a-text-workbook. If this issue is of interest to you, look into it further. 5) The Clinton administration, in a "Draft Comprehensive National Energy Strategy" (February 1998) talks about America's oil as being "abundant," (pg. 4) and it advocates "promoting increased domestic oil ... production" (pg. 2) to reverse this downward trend in U. The peak of the Hubbert Curve of oil production in the U Decimals, Grade 5 Workbook www.patricioginelsa.com. 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[<< wikibooks] LMIs in Control/pages/TDSIC
== The System ==
The problem is to check the stability of the following linear time-delay system

{

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{\displaystyle {\begin{aligned}{\begin{cases}{\dot {x}}(t)&=Ax(t)+A_{d}x(t-d)\\x(t)&=\phi (t),t\in [-d,0],0
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{\displaystyle P>0}

[

A

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{\displaystyle {\begin{bmatrix}A^{T}P+PA+S&PA_{d}\\A_{d}^{T}P&-S\end{bmatrix}}}

<
0

{\displaystyle {\begin{aligned}<0\end{aligned}}}
This LMI has been derived from the Lyapunov Function for the system.
By Schur Complement we can see that the above matrix inequality is equivalent to the Riccati Inequality

A

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{\displaystyle A^{T}P+PA+PA_{d}S^{-1}A_{d}^{T}P+S<0}

== Conclusion: ==
We can now implement these LMIs to do stability analysis for a Time delay system on the delay independent condition

== Implementation ==
The implementation of the above LMI can be seen here
https://github.com/yashgvd/LMI_wikibooks

== Related LMIs ==
Time Delay systems (Delay Dependent Condition)

[1] - LMI in Control Systems Analysis, Design and Applications
LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
D. d. S. Madeira and J. Adamy, "Static output feedback: An LMI condition for stabilizability based on passivity indices," 2016 IEEE Conference on Control Applications (CCA), Buenos Aires, 2016, pp. 960-965.

== Return to Main Page: ==


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 2021-04-16, 23:27 #23 Xyzzy     "Mike" Aug 2002 2·13·313 Posts 1× https://seasonic.com/prime-tx https://www.tomshardware.com/reviews...-psu,5510.html https://www.kitguru.net/components/p...supply-review/ $338.33 including shipping and taxes. Attached Thumbnails 2021-04-16, 23:31 #24 masser Jul 2003 wear a mask 7×229 Posts Quote:  Originally Posted by Xyzzy https://seasonic.com/prime-tx https://www.tomshardware.com/reviews...-psu,5510.html https://www.kitguru.net/components/p...supply-review/$338.33 including shipping and taxes.
:drool:

 2021-04-16, 23:39 #25 Xyzzy     "Mike" Aug 2002 2·13·313 Posts Attached Thumbnails
2021-04-17, 07:52   #26
LaurV
Romulan Interpreter

Jun 2011
Thailand

25·5·59 Posts

Quote:
 Originally Posted by Xyzzy Code: Model name: Intel(R) Core(TM) i9-10980XE CPU @ 3.00GHz CPU MHz: 3000.000 CPU max MHz: 4800.0000 CPU min MHz: 1200.0000 ... Code: $cat results.bench.txt | grep 6144 ... Timings for 6144K FFT length (18 cores, 1 worker): 1.88 ms. Throughput: 531.20 iter/sec. Quote:  Originally Posted by Xyzzy TL;DR - This CPU is a tiny bit faster than a Vega 64 GPU for PRP work. Code: [Main thread Apr 10 20:26] Mersenne number primality test program version 29.8 [Main thread Apr 10 20:26] Optimizing for CPU architecture: Core i3/i5/i7, L2 cache size: 18x1 MB, L3 cache size: 25344 KB [Work thread Apr 10 20:26] Starting Gerbicz error-checking PRP test of M77936863 using AVX-512 FFT length 4200K, Pass1=1920, Pass2=2240, clm=1, 18 threads [Work thread Apr 10 20:28] Iteration: 100000 / 77936863 [0.12%], ms/iter: 1.124, ETA: 24:17:50 ... [Work thread Apr 10 20:45] Iteration: 1000000 / 77936863 [1.28%], ms/iter: 1.122, ETA: 23:58:56 [Work thread Apr 10 20:45] Gerbicz error check passed at iteration 1000000. Quote:  During this test the CPU runs at 2.8GHz all-core (AVX-512) and uses ~200W. The temps stabilize at 57°C with a Noctua UH-12S HSF with the computer out in the open on a test bench. Same here, we got the same ballpark, more or less, except we stay below 48°C with one of these. Not an endorsement, we bought the cheapest we found that fits for 2066 socket. We think an x120 is too small for the 18-cores, ~200W monster (we try to stay in 165W by limiting the power from mobo) and we do not have space in the case for the x360 or larger. The x240 will fit perfectly there. We will have to put another "two times x120" and "two times x240" in the box, from the 4 GPU cards (only one 1080Ti with an x120 cooler mounted in the picture, which drives the monitor, but the final build will have 2x1080Ti and 2x2080Ti, mix cooling, i.e. they have both water and air cooling, as seen in the photo attached below, and in other former pictures we posted here around in the past). (and DON'T make fun of my barbeque stick! That's an important piece of hardware, without it the card would fall on a side, as there is no fixing frame nor screws, and it is quite heavy. we don't want any magic smoke getting out from any component also, there is one 1T 2280 M.2 under the wooden stick, covered by the chipset's cooling block, you can see it if you look careful to the photo; there is no other ssd or hdd connected; no connection to the subject, just boasting ). Initially (before tuning the clocks and FFTs), the lower FFTs were a bit faster, but the larger were slower (capsized at 501 iter/second), compared with your benchmarks. Note that we decided to go for 64GB RAM at 3200 - we wanted 128 initially, possible higher clocks, but the price and especially the lead-time were lousy, and we concluded that for the current (old) rig we "never" used all the ram (i7-6950x, 128G at 2400), so we bailed out at 64 for the new one, which we could buy directly at the counter - advantage of buying it locally is that, in case of failure, we can go back and throw it in their heads (joking, people here at local "JIB computers" were always very nice and replaced all our bad parts, when they went into the weeds, which didn't happen too often, but it did happen sometimes (grrr... wanted to write "now and then" but decided against using "now" and started knocking on head wood ). Quote:  Originally Posted by Xyzzy We have a W10 "Home" license but we were unable to upgrade it to a "Pro" license. When you get win10 (create bootable usb, directly from M$ site) there is no difference between home or pro. Only enterprise is different. If you installed and registered as home, there is a way to "upgrade" to pro, but it is complicate, requires deleting files, etc. OTOH, unless you really do "strange things" in your computer, you won't need pro, you can do everything you want with home. Look to the comparison list, the difference is much smaller than it was for former windozes (like between win7 home vs pro), and the price tells it too (like 139 vs 199 bucks). Possibly, install anydesk on top to compensate for missing network features, and you are good to go. Performance-wise, there is no difference (we tested that, be sure!).

We are here right now in the same situation, we tried without success to install win 7 (from original dvd, paid many years ago to M$) but the mobo is too new and it crashes somewhere in the middle of the installation process, in fact, it doesn't crash, but the mouse and keyboard are inaccessible, and we can't click a "next" button in spite of the fact that we reconnected them to different USB ports, and the animation on screen is still running (so the computer waits for us to click next, or "tab" into it, but neither the keyboard nor the mouse works). We assume Gigabyte, Microsoft and Intel conspired against us, they have a section of micro-code hidden in Bios, or wherever, which says "if the user is LaurV and the time zone is Thailand and the installation time is between 1:00 AM and 4:00 AM, then do not install win 7, and lock the keyboard and the mouse". Therefore, we had to dld win10 installer from M$, and create the bootable usb and install win 10. The download, as well as installation, went unbelievable smooth! We didn't register it (as either home or pro) yet. We are still "deciding" about the fact that we can use this opportunity to install some Linux distro and start the funk once for all learn some real operating system (for long an item in our "todo list", but never found the time, motivation, cleverness, etc., therefore we still suck at that subject). But as we were for 40 years a windoze user (and programmer, we ported WinCE and some Embed versions to many devices that the factory we work for, produced along the years), we continue to procrastinate that task...

Ok, and now, 521 restarts later (or may be 607 restarts, we are nor sure) we got something like this (edited to look similar to yours, the details in the attached zip).

We would be interested if you "tune" yours in the same fashion (try all FFT combinations, with 1, 2, 3, 6, 9 workers, these 18 cores have strange habits when marry each-other...)
Code:
Core/Workers equal split:

FFTlen=6144K, Type=3, Arch=8, Pass1=768,  Pass2=8192, clm=1 (18 cores, 1 worker) :  1.82 ms.  Throughput: 549.85 iter/sec.
FFTlen=6144K, Type=3, Arch=8, Pass1=1024, Pass2=6144, clm=1 (18 cores, 2 workers):  5.32,  5.56 ms.  Throughput: 368.02 iter/sec.
FFTlen=6144K, Type=3, Arch=8, Pass1=768,  Pass2=8192, clm=2 (18 cores, 3 workers):  8.85,  8.81,  8.82 ms.  Throughput: 339.84 iter/sec.
FFTlen=6144K, Type=3, Arch=8, Pass1=768,  Pass2=8192, clm=2 (18 cores, 6 workers): 18.93, 18.58, 18.22, 18.67, 18.76, 18.33 ms.  Throughput: 322.98 iter/sec.
FFTlen=6144K, Type=3, Arch=8, Pass1=768,  Pass2=8192, clm=2 (18 cores, 9 workers): 28.39, 28.30, 28.28, 28.34, 27.70, 28.13, 27.91, 28.30, 27.88 ms.  Throughput: 319.88 iter/sec.

Xyzzy's test (best, after tuning):

FFTlen=6144K, Type=3, Arch=8, Pass1=768,  Pass2=8192, clm=2 (1 core, 1 worker):   19.38 ms.  Throughput: 51.60 iter/sec.
FFTlen=6144K, Type=3, Arch=8, Pass1=768,  Pass2=8192, clm=2 (2 cores, 1 worker):   9.94 ms.  Throughput: 100.56 iter/sec.
FFTlen=6144K, Type=3, Arch=8, Pass1=768,  Pass2=8192, clm=2 (3 cores, 1 worker):   6.68 ms.  Throughput: 149.63 iter/sec.
FFTlen=6144K, Type=3, Arch=8, Pass1=768,  Pass2=8192, clm=2 (4 cores, 1 worker):   5.07 ms.  Throughput: 197.42 iter/sec.
FFTlen=6144K, Type=3, Arch=8, Pass1=1024, Pass2=6144, clm=2 (5 cores, 1 worker):   4.13 ms.  Throughput: 241.90 iter/sec.
FFTlen=6144K, Type=3, Arch=8, Pass1=1024, Pass2=6144, clm=2 (6 cores, 1 worker):   3.49 ms.  Throughput: 286.43 iter/sec.
FFTlen=6144K, Type=3, Arch=8, Pass1=1024, Pass2=6144, clm=2 (7 cores, 1 worker):   3.03 ms.  Throughput: 329.80 iter/sec.
FFTlen=6144K, Type=3, Arch=8, Pass1=768,  Pass2=8192, clm=2 (8 cores, 1 worker):   2.77 ms.  Throughput: 361.65 iter/sec.
FFTlen=6144K, Type=3, Arch=8, Pass1=768,  Pass2=8192, clm=2 (9 cores, 1 worker):   2.54 ms.  Throughput: 393.35 iter/sec.
FFTlen=6144K, Type=3, Arch=8, Pass1=768,  Pass2=8192, clm=1 (10 cores, 1 worker):  2.39 ms.  Throughput: 418.04 iter/sec.
FFTlen=6144K, Type=3, Arch=8, Pass1=768,  Pass2=8192, clm=2 (11 cores, 1 worker):  2.28 ms.  Throughput: 438.67 iter/sec.
FFTlen=6144K, Type=3, Arch=8, Pass1=1024, Pass2=6144, clm=1 (12 cores, 1 worker):  2.17 ms.  Throughput: 460.50 iter/sec.
FFTlen=6144K, Type=3, Arch=8, Pass1=768,  Pass2=8192, clm=1 (13 cores, 1 worker):  2.11 ms.  Throughput: 473.72 iter/sec.
FFTlen=6144K, Type=3, Arch=8, Pass1=1024, Pass2=6144, clm=1 (14 cores, 1 worker):  2.04 ms.  Throughput: 490.41 iter/sec.
FFTlen=6144K, Type=3, Arch=8, Pass1=1024, Pass2=6144, clm=1 (15 cores, 1 worker):  1.98 ms.  Throughput: 506.07 iter/sec.
FFTlen=6144K, Type=3, Arch=8, Pass1=1024, Pass2=6144, clm=1 (16 cores, 1 worker):  1.93 ms.  Throughput: 517.92 iter/sec.
FFTlen=6144K, Type=3, Arch=8, Pass1=768,  Pass2=8192, clm=1 (17 cores, 1 worker):  1.90 ms.  Throughput: 527.26 iter/sec.
FFTlen=6144K, Type=3, Arch=8, Pass1=768,  Pass2=8192, clm=1 (18 cores, 1 worker):  1.86 ms.  Throughput: 536.87 iter/sec.

Front line tests (tuning):

FFTlen=5600K, Type=3, Arch=8, Pass1=896,  Pass2=6400,  clm=4 (18 cores, 1 worker):  2.06 ms.  Throughput: 485.11 iter/sec.
FFTlen=5600K, Type=3, Arch=8, Pass1=896,  Pass2=6400,  clm=2 (18 cores, 1 worker):  1.56 ms.  Throughput: 641.92 iter/sec.
FFTlen=5600K, Type=3, Arch=8, Pass1=896,  Pass2=6400,  clm=1 (18 cores, 1 worker):  1.53 ms.  Throughput: 654.65 iter/sec.
FFTlen=5600K, Type=3, Arch=8, Pass1=1280, Pass2=4480,  clm=2 (18 cores, 1 worker):  1.75 ms.  Throughput: 572.56 iter/sec.
FFTlen=5600K, Type=3, Arch=8, Pass1=1280, Pass2=4480,  clm=1 (18 cores, 1 worker):  1.56 ms.  Throughput: 639.73 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=192,  Pass2=30720, clm=4 (18 cores, 1 worker):  2.05 ms.  Throughput: 487.83 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=192,  Pass2=30720, clm=2 (18 cores, 1 worker):  2.51 ms.  Throughput: 397.75 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=192,  Pass2=30720, clm=1 (18 cores, 1 worker):  5.91 ms.  Throughput: 169.20 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=640,  Pass2=9216,  clm=4 (18 cores, 1 worker):  1.84 ms.  Throughput: 543.78 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=640,  Pass2=9216,  clm=2 (18 cores, 1 worker):  1.64 ms.  Throughput: 608.70 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=640,  Pass2=9216,  clm=1 (18 cores, 1 worker):  1.59 ms.  Throughput: 627.25 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=768,  Pass2=7680,  clm=4 (18 cores, 1 worker):  2.00 ms.  Throughput: 499.62 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=768,  Pass2=7680,  clm=2 (18 cores, 1 worker):  1.69 ms.  Throughput: 591.33 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=768,  Pass2=7680,  clm=1 (18 cores, 1 worker):  1.62 ms.  Throughput: 618.17 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=960,  Pass2=6144,  clm=4 (18 cores, 1 worker):  2.16 ms.  Throughput: 462.71 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=960,  Pass2=6144,  clm=2 (18 cores, 1 worker):  1.67 ms.  Throughput: 600.49 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=960,  Pass2=6144,  clm=1 (18 cores, 1 worker):  1.59 ms.  Throughput: 629.88 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=1152, Pass2=5120,  clm=2 (18 cores, 1 worker):  1.74 ms.  Throughput: 573.79 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=1152, Pass2=5120,  clm=1 (18 cores, 1 worker):  1.60 ms.  Throughput: 625.71 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=1280, Pass2=4608,  clm=2 (18 cores, 1 worker):  1.84 ms.  Throughput: 542.17 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=1280, Pass2=4608,  clm=1 (18 cores, 1 worker):  1.67 ms.  Throughput: 599.78 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=1536, Pass2=3840,  clm=2 (18 cores, 1 worker):  2.02 ms.  Throughput: 494.24 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=1536, Pass2=3840,  clm=1 (18 cores, 1 worker):  1.72 ms.  Throughput: 581.54 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=1920, Pass2=3072,  clm=2 (18 cores, 1 worker):  2.21 ms.  Throughput: 452.96 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=1920, Pass2=3072,  clm=1 (18 cores, 1 worker):  1.73 ms.  Throughput: 576.57 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=2304, Pass2=2560,  clm=1 (18 cores, 1 worker):  1.84 ms.  Throughput: 544.70 iter/sec.
FFTlen=5760K, Type=3, Arch=8, Pass1=3072, Pass2=1920,  clm=1 (18 cores, 1 worker):  2.50 ms.  Throughput: 400.35 iter/sec.
Attached Files
 bench.zip (21.5 KB, 7 views)

Last fiddled with by LaurV on 2021-04-17 at 13:41 Reason: add links, fix typos, line spacing, alignment in code tags

 2021-04-17, 12:46 #27 Xyzzy     "Mike" Aug 2002 2×13×313 Posts We have a BBQ stick like yours that we use to push on the slot tab to remove the video card. Our fingers are too fat to do it manually. We only went with W10 "Pro" because "W10" Home doesn't support 256GB of memory. We will tune the CPU once we get a more robust cooling system installed.
2021-04-18, 05:44   #28
LaurV
Romulan Interpreter

Jun 2011
Thailand

25×5×59 Posts

About the 256G, if this is your concern only, did you try "edu"? There is no memory restriction (well, the 2TB of the 40 bits addresses, but that should be enough), and you may get it free. Some people say "edu pro" is better than "pro", and "edu" (based on "enterprise") is even better. Unless you already bought "pro"...

A friend of mine who read my post above, where I said that "pro" and "home" are the same fast said that may not be true, and "home" may actually be about 1% to 3% faster. The reason may be the WIP (Windows information protection). I don't know what WIP does, but the feature comparison list says that:

Quote:
 WIP helps to protect against potential data leakage without otherwise interfering with the employee experience. WIP also helps to protect enterprise apps and data against accidental data leaks on enterprise-owned devices and personal devices that employees bring to work, without requiring changes to your environment or other apps.
People interpret that like a "patch" for the "meltdown" and "spectre" (they wrote special code to carefully "arrange" the data in memory to avoid the side-leaks specific to these two), which is active on "pro" (checkboxed in the list) but not on "home", and the patch is known from the past (Intel microcode for CPUs) to make everything in the system about 1 to 3 percent slower. OTOH, my friend says he never tested it to check if it is true. It may be only gossips.

Related to thermals, one (BIG) issue with the water cooled CPU in the open (i.e. no case fans) is the fact that, well, there are no case fans...

While the CPU temperature stays at 47°C, or 48°C max (anyhow, under 50°C, or for Americans 122°F ), the memories get close to 60°C (140°F), and the VRM mosfets together with the big iron backplate behind the mobo get close to 70°C (158°F). That because, when you use the air cooler, there is a lot of air moving around, cold, warm, or hot, but it is moving, taking away with it the heat from the memories, chipset, mosfets, and other things which haven't their own fans. When everything it "on water", and no case walls to hung the radiators, there is no air blown anywhere near the mobo, chipset, power mosfets, nothing to dissipate that heat, so they really DO get hot.

For my build, with 2+2+2+1+1 x120 radiator fans, a lot of study will need to be done related to the direction of blowing for the fans. Inside or outside of the case? This will involve many hours of tests. No joke. Probably a mix of them will be the best, but how? (yes, fans blowing "hot" air into the case! This was not a typing mistake. Do you guess why?)

Last fiddled with by LaurV on 2021-04-18 at 06:08 Reason: spacing, typos

 2021-04-18, 13:06 #29 Xyzzy     "Mike" Aug 2002 2×13×313 Posts We don't consider 70°C for VRMs to be too hot. Our board has two miniature VRM fans. The "HS Fan" reading in the second attachment is the VRM fan speed. Attached Thumbnails
2021-04-18, 13:07   #30
Xyzzy

"Mike"
Aug 2002

2×13×313 Posts

Quote:
 Originally Posted by LaurV (yes, fans blowing "hot" air into the case! This was not a typing mistake. Do you guess why?)
The "hot" air from the radiators will only be a few degrees above ambient and will also be much cooler than the rest of the heated computer parts?

 2021-04-18, 15:47 #31 LaurV Romulan Interpreter     Jun 2011 Thailand 25·5·59 Posts Yep, that's half of the solution. More exactly, the second half The idea is that "heating" and "cooling" is (at molecular level) just thermal agitation (or respective, stopping it). Fast molecules (hot) hit slow molecules (cold) and lose (transfer) part of their energy. More cold molecules, more heat transfer. Why the water (and liquids in general) are better coolers than air (and gases in general)? Because liquids are more "dense", they contain a lot more molecules per unit of volume (say, cubic centimeter), so, more bumps, better energy (heat) transfer. That works for air too, and for all compressible fluids in general, the compressed fluid always cools (or heats) better than the rarefied one. More cold-air molecules hit your radiator wings, faster they will take the heat away. If you put your mobo inside a 10-bar air chamber it will cool faster. In fact, if you put it in absolute vacuum, it will never cool, because there is no air to take the heat away. The mobo will heat to hundreds or thousands of degrees, when, if it will not melt, it will start losing more energy by IR radiation, which can propagate through vacuum. Haha. I won many bets in my life by proving to friends and colleagues (who dared to call themselves "IT Professionals") that if you put all your case fans to blow air into the case, and let the air go out through "special" holes, to create the right air flow inside, then the box stays cooler, compared to the "classic" method where some fans are blowing cold air "into" the case (usually in front, or below the housing, cold air is pushed into the case) and some other fans are blowing air out of the case, "to remove the heat" (usually in the back of the house, to avoid hot air blowing "into your face", or on the top of the house, because there the air "is hotter", because "the hot air moves to the top", which is by itself false, the air moves so fast that it doesn't have time to "accumulate" the heat in some corner or top of the housing, even with just a single fan). When you have the fans pushing more air into the case, the pressure inside increase, even if only a bit (because the case has many holes). The best solution would be to have the CPU and GPU fans to blow the hot air OUT of the case, or mount them outside so they would not influence the air movement inside of the case, but then, have enough additional case fans to blow enough cold air from outside INTO the case, to have higher pressure inside. This not only cools faster, but also avoids condensation inside. Unfortunately, for my build that won't be possible, I can not mount the fans outside, because all the hoses for the "all in one" coolers are short and unmovable (i.e. it would need to scrap them all and make a classic water loop, a lot of effort and money), and if I mount them on the case's walls oriented in such a way to "blow the hot air out of the house", then eight such strong fans will "vacuum" the case inside (i.e. decrease the pressure), slowing down the cooling of the passive stuff (chipset, mosfets), and moreover, I will not have much space there to put more fans to bring more air in (beside of a 28 cm fan lateral, and a 14 cm in front, all walls are "taken"). Mosfets, RAM, and chipset must have some fan to blow cold air ONTO them. Like on your mobo (which is brilliant, by the way!). So, at least some of the 8 fans will have to blow towards inside. This is what I assume, but I may be wrong. Therefore, it is a quest between "taking the air out and decreasing the pressure", or "putting warm air in", both with negative effects for cooling, and the effects can not be quantified unless trying many possibilities. It will all depend on the how warm is the air that I put inside, and how much is the pressure difference (i.e. how much "compressed air flow" I can put inside. Probably, the CPU fans will blow inside, because the CPU circuit will be the "colder". If the CPU can stay in (say) the 60s, then the water temperature will stay somewhere in the 50s (thermal resistance of the CPU cover and the copper block of the cooler/pump), which means that the air will be in the 40s (thermal resistance of the aluminum reservoir, yeah, I know, don't ever talk to me about two different metals in the circuit, didn't I say this is a cheap cooler?), as long as the temperature in the room stays around 30 (in hot days). The water runs very fast, in fact the water that comes out from the water blocks is just 1-2-3 degrees hotter that the water that enters water blocks, exactly as you said, the water has no time to "get hot" there. In average the water is about 10-15 degrees colder than the CPU (otherwise, no cooling, the thermal resistance of the block works like the resistance in electronics, except it is thermal ). Similar, the water that comes out of the radiator is just 1-2-3 degres colder than the water that enters the radiator. Same for the air, which is in average 10-12 degrees colder than the water (for a x240 radiator), (otherwise no cooling), the air that is sucked by the fans is at room temperature, and the air that is blown by the same fans through the radiator's mesh is just few degrees hotter. No time to get "hot". So, blowing it into the case may help with cooling due to pressurizing, or may not, due to the fact that is warmer. We will see... Last fiddled with by LaurV on 2021-04-18 at 15:56
 2021-04-18, 18:37 #32 Nick     Dec 2012 The Netherlands 68916 Posts At home with LaurV...
 2021-04-18, 23:04 #33 ewmayer ∂2ω=0     Sep 2002 República de California 7·11·151 Posts Re. Hot-air-blowing: [hot by human standards] != [hot by chipset standards]. The 2nd Radeon 7 GPU I added to my old ATX-case Haswell system (all case fans long-ago kaput, ventilation all via ambient air thanks to removed case side panels) last summer sits with its bottom a mere ~3cm above the hot-air vent fan of the PSU, so roughly half the GPU intake air is preheated by the PSU. Said GPU2 still runs - at same settings - roughly 10C cooler than the one, GPU1, sitting above it. Part of that is of course because GPU1 is also getting air partly prewarmed by GPU2, but the main thing is that air at (roughly 50C) blowing from PSU directly into the intake fans of GPU2 has a significant cooling effect relative to the hotter-running GPU.

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# How do we prove that something is unprovable?

I have read somewhere there are some theorems that are shown to be "unprovable". It was a while ago and I don't remember the details, and I suspect that this question might be the result of a total misunderstanding. By the way, I assume that unprovable theorem does exist. Please correct me if I am wrong and skip reading the rest.

As far as I know, the mathematical statements are categorized into: undefined concepts, definitions, axioms, conjectures, lemmas and theorems. There might be some other types that I am not aware of as an amateur math learner. In this categorization, an axiom is something that cannot be built upon other things and it is too obvious to be proved (is it?). So axioms are unprovable. A theorem or lemma is actually a conjecture that has been proved. So "a theorem that cannot be proved" sounds like a paradox.

I know that there are some statements that cannot be proved simply because they are wrong. I am not addressing them because they are not theorems. So what does it mean that a theorem is unprovable? Does it mean that it cannot be proved by current mathematical tools and it may be proved in the future by more advanced tools that are not discovered yet? So why don't we call it a conjecture? If it cannot be proved at all, then it is better to call it an axiom.

Another question is, how can we be sure that a theorem cannot be proved? I am assuming the description might be some high level logic that is way above my understanding. So I would appreciate if you put it into simple words.

Edit- Thanks to a comment by @user21820 I just read two other interesting posts, this and this that are relevant to this question. I recommend everyone to take a look at them as well.

• I was just thinking of preparing a talk on this very subject... Nov 23 '16 at 12:06
• "In this categorization, an axiom is something that cannot be built upon other things and it is too obvious to be proved (is it?). So axioms are unprovable. " Not at all. Often axioms are found to be redundant: Hilbert's axiom "$a \implies a$" was found to be redundant (in other words, it was provable from other axioms). Happens all the time. Nov 23 '16 at 12:32
• While logic is a rich subject, it tends to be very shallow in the parts where laypeople think there is depth (e.g. the notion of "axiom"). Most of the ideas you have about logic are not actually about logic, but instead ideas about philosophy or pedagogy
– user14972
Nov 23 '16 at 15:18
• Stefan's answer has identified that the main issue is your understanding of "unprovable", which is always relative. See math.stackexchange.com/a/1643073 for a brief outline of some points I wish to highlight, and math.stackexchange.com/a/1808558 for a little more on foundational aspects. Nov 25 '16 at 7:02
• 'Theorem" has 2 different meanings. In the usual language, it means "a proposition (of some importance) for which someone has given have an acceptable proof". It is a result obtained by people. In logic, it is a well-formed formula which happens to be true in some theory. When talking about "provability", you are in the second context; so theorem exists even without written proof, without the exsitence of proof, - and even without anybody stating them. Nov 25 '16 at 10:01

When we say that a statement is 'unprovable', we mean that it is unprovable from the axioms of a particular theory.

Here's a nice concrete example. Euclid's Elements, the prototypical example of axiomatic mathematics, begins by stating the following five axioms:

Any two points can be joined by a straight line

Any finite straight line segment can be extended to form an infinite straight line.

For any point $$P$$ and choice of radius $$r$$ we can form a circle centred at $$P$$ of radius $$r$$

All right angles are equal to one another.

[The parallel postulate:] If $$L$$ is a straight line and $$P$$ is a point not on the line $$L$$ then there is at most one line $$L'$$ that passes through $$P$$ and is parallel to $$L$$.

Euclid proceeds to derive much of classical plane geometry from these five axioms. This is an important point. After these axioms have been stated, Euclid makes no further appeal to our natural intuition for the concepts of 'line', 'point' and 'angle', but only gives proofs that can be deduced from the five axioms alone.

It is conceivable that you could come up with your own theory with 'points' and 'lines' that do not resemble points and lines at all. But if you could show that your 'points' and 'lines' obey the five axioms of Euclid, then you could interpret all of his theorems in your new theory.

In the two thousand years following the publication of the Elements, one major question that arose was: do we need the fifth axiom? The fifth axiom - known as the parallel postulate - seems less intuitively obvious than the other four: if we could find a way of deducing the fifth axiom from the first four then it would become superfluous and we could leave it out.

Mathematicians tried for millennia to find a way of deducing the parallel postulate from the first four axioms (and I'm sure there are cranks who are still trying to do so now), but were unable to. Gradually, they started to get the feeling that it might be impossible to prove the parallel postulate from the first four axioms. But how do you prove that something is unprovable?

The right approach was found independently by Lobachevsky and Bolyai (and possibly Gauss) in the nineteenth century. They took the first four axioms and replaced the fifth with the following:

[Hyperbolic parallel postulate:] If $$L$$ is a straight line and $$P$$ is a point not on the line $$L$$ then there are at least two lines that pass through $$P$$ and are parallel to $$L$$.

This axiom is clearly incompatible with the original parallel postulate. The remarkable thing is that there is a geometrical theory in which the first four axioms and the modified parallel postulate are true.

The theory is called hyperbolic geometry and it deals with points and lines inscribed on the surface of a hyperboloid:

In the bottom right of the image above, you can see a pair of hyperbolic parallel lines. Notice that they diverge from one another.

The first four axioms hold (and you can check this), but now if $$L$$ is a line and $$P$$ is a point not on $$L$$ then there are infinitely many lines parallel to $$L$$ passing through $$P$$. So the original parallel postulate does not hold.

This now allows us to prove very quickly that it is impossible to prove the parallel postulate from the other four axioms: indeed, suppose there were such a proof. Since the first four axioms are true in hyperbolic geometry, our proof would induce a proof of the parallel postulate in the setting of hyperbolic geometry. But the parallel postulate is not true in hyperbolic geometry, so this is absurd.

This is a major method for showing that statements are unprovable in various theories. Indeed, a theorem of Gödel (Gödel's completeness theorem) tells us that if a statement $$s$$ in the language of some axiomatic theory $$\mathbb T$$ is unprovable then there is always some structure that satisfies the axioms of $$\mathbb T$$ in which $$s$$ is false. So showing that $$s$$ is unprovable often amounts to finding such a structure.

It is also possible to show that things are unprovable using a direct combinatorial argument on the axioms and deduction rules you are allowed in your logic. I won't go into that here.

You're probably interested in things like Gödel's incompleteness theorem, that say that there are statements that are unprovable in a particular theory called ZFC set theory, which is often used as the foundation of all mathematics (note: there is in fact plenty of mathematics that cannot be expressed in ZFC, so all isn't really correct here). This situation is not at all different from the geometrical example I gave above:

If a particular statement is neither provable nor disprovable from the axioms of all mathematics it means that there are two structures out there, both of which interpret the axioms of all mathematics, in one of which the statement is true and in the other of which the statement is false.

Sometimes we have explicit examples: one important problem at the turn of the century was the Continuum Hypothesis. The problem was solved in two steps:

• Gödel gave a structure satisfying the axioms of ZFC set theory in which the Continuum Hypothesis was true.
• Later, Cohen gave a structure satisfying the axioms of ZFC set theory in which the Continuum Hypothesis was false.

Between them, these results show that the Continuum Hypothesis is in fact neither provable nor disprovable in ZFC set theory.

• Nice answer. Do you want to add a few words about the third version of the parallel postulate, in which there are no parallel lines? Nov 23 '16 at 16:54
• I learned a lot from all answers and I can't resist to emphasize that marking this answer as accepted doesn't imply undervaluing other ones Nov 24 '16 at 7:08
• It isn't quite true that Euclid proved his theorems from those 5 axioms alone. Over the years, mathematicians have identified several other assumptions he made without realizing it. These additional assumptions are also axioms of Euclidean geometry. One such assumption is that a line separates a plane into two halves, such that any line connecting points in one half with points in the other must intersect the first line. Nov 24 '16 at 14:44
• @HenningMakholm SGA/EGA make heavy use of Grothendieck universes, which can't be shown to exist in ZFC alone. Now, much of the material in SGA/EGA can be developed in ZFC, but the convenience of the Grothendieck universes approach makes me uneasy of claiming that ZFC is a foundation for all mathematics. Nov 24 '16 at 16:28
• @JohnGowers: Grothendieck universes don’t need to be viewed as outside or beyond ZFC — they can reasonably be seen as just an extra existence assumption, the basic framework still remains ZFC. I work in type theory and constructive maths, so I’m a big proponent of alternatives, but I don’t think there’s any problem in saying that ZFC’s a satisfactory foundation for the vast majority of current mathematics (including EGA, SGA etc). Nov 28 '16 at 11:22

First of all in the following answer I allowed myself (contrary to my general nature) to focus my efforts on simplicity, rather than formal correctness.

In general, I think that the way we teach the concept of axioms is rather unfortunate. While traditionally axioms were thought of as statements that are - in some philosophical way - obviously true and don't need further justifications, this view has shifted a lot in the last century or so. Rather than thinking of axioms as obvious truths think of them as statements that we declare to be true. Let $\mathcal A$ be a set of axioms. We can now ask a bunch of questions about $\mathcal A$.

• Is $\mathcal A$ self-contradictory? I.e. does there exist a proof (<- this needs to be formalized, but for the sake of simplicity just think of your informal notion of proofs) - starting from formulas in $\mathcal A$ that leads to a contradiction? If that's the case, then $\mathcal A$ was poorly chosen. If all the statements in $\mathcal A$ should be true (in a philosophical sense), then they cannot lead to a contradiction. So our first requirement is that $\mathcal A$ - should it represent a collection of true statements - is not self-contradictory.
• Does $\mathcal A$ prove interesting statements? Take for example $\mathcal A$ as the axioms of set theory (e.g. $\mathcal A = \operatorname{ZFC}$). In this case we can prove all sorts of interesting mathematical statements. In fact, it seems reasonable that every mathematical theorem that can be proved by the usual style of informal proofs, can be formally proved from $\mathcal A$. This is one of the reasons, the axioms of set theory have been so successful.
• Is $\mathcal A$ a natural set of axioms? ...
• ...
• Is there a statement $\phi$ which $\mathcal A$ does not decide? I.e. is there a statement $\phi$ such that there is no proof of $\phi$ or $\neg \phi$ starting from $\mathcal A$?

The last point is what we mean when we say that $\phi$ is unprovable from $\mathcal A$. And if $\mathcal A$ is our background theory, say $\mathcal A = \operatorname{ZFC}$, we just say that $\phi$ is unprovable.

By a very general theorem of Kurt Gödel, any natural set of axioms $\mathcal A$ has statements that are unprovable from it. In fact, the statement "$\mathcal A$ is not self-contradictory" is not provable from $\mathcal A$. So, while natural sets of axioms $\mathcal A$ are not self-contradictory - they themselves cannot prove this fact. This is rather unfortunate and demonstrates that David Hilbert's program on the foundation of mathematics - in its original form - is impossible. The natural workaround is something contrary to the general nature of mathematics - a leap of faith: If $\mathcal A$ is a sufficiently natural set of axioms (or otherwise certified), we believe that it is consistent (or - if you're more like me - you assume it is consistent until you see a reason not to).

This is - for example - the case for $\mathcal A = \operatorname{ZFC}$ and for the remainder of my answer, I will restrict myself to this scenario. Now that we know that $\mathcal A$ does not decide all statements (and arguably does not prove some true statements - like its consistency), a new question arises:

• Does $\operatorname{ZFC}$ decide all mathematical statements? In other words: Is there a question about typical mathematical objects that $\operatorname{ZFC}$ does not answer?

The - to some people unfortunate - answer is yes and the most famous example is

$\operatorname{ZFC}$ does not decide how many real numbers there are.

Actually proving this fact, took mathematicians (logicians) many decades. At the end of this effort, however, we not only had a way to prove this single statement, but we actually obtained a very general method to prove the independence of many statements (the so-called method of forcing, introduced by Paul Cohen in 1963).

The idea - roughly speaking - is as follows: Let $\phi$ be a statement, say

$\phi \equiv$ "there is no infinity strictly between the infinity of $\mathbb N$ and of $\mathbb R$"

Let $\mathcal M$ be a model of $\operatorname{ZFC}$. Starting from $\mathcal M$ we would like to construct new models $\mathcal M_{\phi}$ and $\mathcal M_{\neg \phi}$ of $\operatorname{ZFC}$ such that $\mathcal M_{\phi} \models \phi$ and $\mathcal M_{\neg \phi} \models \neg \phi$ (i.e. $\phi$ is true in $\mathcal M_{\phi}$ and $\phi$ is false in $\mathcal M_{\neg \phi}$). If this is possible, then this proves that $\phi$ is not decided by $\operatorname{ZFC}$. Why is that?

Well, if it were decided by $\operatorname{ZFC}$, then there would be a proof of $\phi$ or a proof of $\neg \phi$. Let us say that $\phi$ has a proof (the other case is the same). Then, by soundness of our proofs, any model that satisfies $\operatorname{ZFC}$ must satisfy $\phi$, so there cannot be a model $\mathcal M_{\neg \phi}$ as above.

• @polfosol Well, because the truth of a given statement - unless implied by your axioms - is debatable. You are free to add this true statement to your axioms (and we regularly do that), but to other people the negation may seem to be true and they might decide to accept that as an axiom. Nov 23 '16 at 12:59
• @polfosol I myself don't consider any statement to be true (in an absolute sense) - at least I think I don't. Rather I work under the reasonable assumption that they are not self-contradictory and try to figure out what their consequences are. Nov 23 '16 at 13:00
• @polfosol Because - thus far - set theory has been most successful (or unfortunate - depending on your point of view) in determining natural, unprovable statements. However, there are other examples. See for example Goodstein's Theorem. Nov 23 '16 at 13:10
• When you consider whether A is a "natural" set of axioms, what do you mean by "natural"? Nov 23 '16 at 14:34
• @Stefan Strictly speaking, other theories have been much more successful than set theory for determining natural unprovable statements. For example, the statement '$G$ is abelian' is undecidable in the theory of groups. The point is that when we say undecidable, we usually mean undecidable in set theory , so the natural examples will also come from set theory. Nov 23 '16 at 15:15

Your question is partially based on an error of terminology. We don't speak of unprovable theorems - as you say, being a theorem implies having a proof. The correct thing to speak of is unprovable statements or unprovable assertions.

That said, it is possible for a statement to be a theorem in one context but not in another, in which case you could say that, in the second context, it is an unprovable theorem. Example: All proofs of the Pythagorean Theorem rely essentially on the Parallel Postulate; so if you try to prove what you can in geometry with the Parallel Postulate omitted (this is called 'absolute geometry'), then the Pythagorean Theorem becomes an unprovable theorem within that context. (this is about the only situation where I think the phrase 'unprovable theorem' is the natural choice)

• +1 and thanks for an easy to follow example of unprovable theorems. Is it worked out somewhere? Mar 15 '17 at 6:52

You have already received some excellent responses, but I just wanted to clarify a few things that I feel may have not received the attention that they require.

For instance, you mentioned

[...] definitions, axioms, conjectures, lemmas and theorems.

It is important to realize that when setting up a mathematical theory, there are really only two things that matter:

1. Statements we define to be true (or, equivalently, false).
2. All other statements, that can either be true or false, or neither; these require a proof.

Definitions and axioms are both a form of type 1. They are quite closely related as axioms usually define some important concept and definitions usually specify that we call $$Z$$ a zork if and only if $$Z$$ has certain properties. For example, consider the statments

Every natural number $$n$$ has a successor $$S(n)$$.

An even number is of the form $$2n$$, for some $$n \in \mathbb N$$.

Are thesee definitions of the successor function and even numbers, respectively? Or are they axioms in a theory about natural numbers?

Similarly the distinction between theorems, conjectures, lemmas etc. is mostly an emotional distinction: they help the reader of a book or article distinguish between the "important" results and the statements that are more like tools and need to be proven only to get an interesting result.

Now on to (un)provable. There are formal definitions of a proof in mathematical logic, but let's not go into them. The main point is that a proof is a logical sequence of steps, starting from the assumptions of a theorem (or lemma, or conjecture, ...), ending with the conclusion of the theorem, and where every step is justified either by an axiom or by a previously proven theorem:

Theorem: (If conditions then) conclusion

Note that the conditions part can be omitted if the conclusion is always true (a tautology).

The "proof" for an axiom is then quite trivial: there are no preconditions and the single step of the proof consists of invoking the axiom.

For a lot of statements, we can usually show that they are true or false, given some set of axioms (and we usually include a set of proven theorems as well that we agree on, for example, you don't invoke Peano's axioms every time you prove something about a natural number). Showing that the theorem is false usually involves finding a counter example, or - more formally - showing that the "anti"theorem

Theorem: (If conditions then) not (conclusion)

is true, again by a proof in which each step is nicely justified.

So for example, we can prove the theorem "6 is even" to be true as follows:

• $$6 = 2 \cdot 3$$
• So $$6 = 2 \cdot n$$ for $$n = 3$$.
• This satisfies the definition of an even number (or, if you have set up things that way: This is an even number by the axiom "Even number").

Similarly we can prove that 7 is not even by showing that there is no number $$n$$ such that $$7 = 2 \cdot n$$ (the only solution is $$n = 3.5$$ and that is not a natural number).

However, suppose that I want to prove "4 is a nortial number". Is that theorem true or false in our simple theory with one axiom about even numbers? No matter what way you invoke the axiom, or any theorem you can prove from it, you cannot show that 4 is a nortial number. However, similarly you cannot prove that "4 is not a nortial number" so the theorem can also not be false. Clearly, your theory does not have enough axioms to be able to reach either conclusion. This is what we mean by "an unprovable statement".

• What is a nortial number? Nov 28 '16 at 9:36
• I have no idea :) Nov 28 '16 at 9:56
• Your answer is pretty good, but like many others, you overlook the importance of "undefined concepts" in definitions. In set theory, what is an element? Does an element exist? What is "membership"? In geometry, we have a choice of a new undefined concept, "sense" referring to clockwise or counterclockwise, or creating a new axiom related to a line partitioning the real projective plane. When you try to define an undefined concept, you generally do so in other terms that lead back to the thing you are trying to define. Nov 29 '16 at 19:31

"Too obvious" doesn't exist in mathematics, we choose to agree about some axioms to build a theory. Axioms mustn't imply each other and mustn't be contradictory.

If a theorem cannot be proven nor denied by the axioms that means it goes beyond boundaries of that theory. So you can join that theorem as another axiom to build more complex theory, or you can join its negation as an axiom to build another theory

In addition, how can be proven that a theorem isn't implied nor denied by axioms?? That would be something like if you don't get contradiction by joining the theorem nor by joining its negation to the set of axioms. Just pass trough whole set of objects you use in the theory... :P

• "we choose to agree about some axioms to build a theory." To be historically accurate, we usually know the theorems we want to prove long before anyone proposes a set of axioms to support them, for almost any theory you will ever find. "Axioms mustn't imply each other and mustn't be contradictory." Axioms often are found to imply each other, but fortunately are not often found to be contradictory. Nov 23 '16 at 13:09
• @DanielV You're right, this was said for an imagined perfectly defined theory. And of course we choose axioms so to satisfy a wanted purpose Nov 23 '16 at 14:51
• Note that In set theory we prove that Zermelo's and axiom of choice are equivenent; each implies the other. Nov 29 '16 at 19:38
• +1 how large will these sets of theorems grow when you keep on joining theorems that are currently beyond the boundaries? Mar 15 '17 at 6:50
• @draks... I guess as large as your imagination reaches Mar 21 '17 at 12:08

A theory is created from building blocks called axioms which are accepted to be true. The axioms need not and cannot be proven (not because they are "too obvious", but because they are independent of the other axioms). Anyway, you have to prove that the set of axioms forming your theory is not contradictory.

Since Gödel, we know that a (rich) theory cannot prove all the logical propositions expressible in its frame, so that some propositions will require extra axioms, hence a stronger theory, to become theorems.

Simplistically speaking, propositions can be shown "unprovable" when their meaning is equivalent to "I am not provable in the frame of the theory". Then if they were provable, they would not be provable, and conversely.

• There are rich and complete theories - take for example the theory of $H_{\omega_{1}}$. The point of Gödel's theorem is, that those cannot be recursively enumerable. Nov 23 '16 at 22:18
• @Stefan: I don't think that this is on the level that the OP expects.
– user65203
Nov 23 '16 at 22:20
• Sure. The reason for my comment is that people often get Gödel's Incompleteness Theorem wrong - which trained me to be very sensitive about its statement (-; Nov 23 '16 at 22:23
• Off topic a bit, but is there a minimal axiomatic structure that contains unprovable statements? is there a maximally complex axiomatic structure that contains NO unprovable statements? Nov 29 '16 at 19:40
• @richard1941 I think you would want to examine philosophy.stackexchange.com/questions/15525/… carefully. And then try to prove maximality or minimality. Since all of the systems I know of are amenable to various axiomatic formulations; you probably need to specify minimal and maximal. Nov 30 '16 at 18:40

It can make sense to call something an "unprovable theorem" if what is meant is that the theorem is unprovable in a given formal system or from a given set of axioms. For instance Gödel's formal statement which informally said "I cannot be proven" is unprovable in the formal system he used, but he also proved it using more powerful, informal tools - thus making it a theorem.

The most common way (AFAIK) to establish $A \not \vdash B$ is to find some statement $X$ such that:

• $A,~X \vdash \lnot B$
• $A \land X$ is consistent

Then you can conclude $A \not \vdash B$ because constructively if $A \vdash B$ then $A ,~ X \vdash B$ then $A ,~ X \vdash B \land \lnot B$ which contradicts the assumption that $A \land X$ are consistent.

For example, if you wanted to show that from "$n$ is divisible by $3$" you cannot prove "$n$ is divisible by $6$" you do the obvious thing of pointing out the $X$ of $n = 9$. Since

• $3|n , ~ n = 9 \vdash \lnot 6|n$
• $(3|n) \land (n = 9)$ is consistent

So $3|n \not \vdash 6|n$. This simple approach can get very complex, mainly because establishing $A \land X$ as being consistent can be extremely demanding, usually an informal model theoretic approach is used for this.

It is also a well known way to establish undecidability, if you can find $X_1$ and $X_2$ such that:

• $A, ~ X_1 \vdash B$
• $A, ~ X_2 \vdash \lnot B$
• $A \land X_1$ is consistent
• $A \land X_2$ is consistent

Then for the same reason $A \not \vdash B$ and $A \not \vdash \lnot B$, so $B$ is independent of $A$.

• There is some circularity in this answer. The claim "$A \land X$ is consistent" is equivalent to claiming the unprovability of inconsistencies. This is unavoidable. Unprovability, like many other things in logic, is never created, only propagated. Oct 12 '18 at 11:12