Is it fair to suggest that the fact a base's symbol which would exist in a higher base but is never truly reflected in the base itself is an example(see below) of incompleteness along the ideas of the theorems? My apologies as I'm mostly self-teaching in these areas and feel I've skipped a lot of interim understanding. I don't know logic notation yet so can't follow any raw work. My example would be as follows;

In binary, base 2, we only ever feature the numbers 0 and 1 in all our numerical representations. Despite the fact it's base 2 the numerical symbol of 2 itself never actually appears in this system as this is instead 10.

Is this an example of the theories of incompleteness? Have I just made a random naive or arbitrary correction or is this a fair conclusion of sorts, if even very simplistic? Thanks in advance.

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    $\begingroup$ "random naive" seems to sum it up pretty well. $\endgroup$ – Gerry Myerson Mar 19 at 11:16
  • $\begingroup$ The word incompleteness,as it is usually used, doesn't have much to do with symbols $\endgroup$ – Max Mar 19 at 11:16
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    $\begingroup$ "Is this an example of the theories of incompleteness?" No. It has nothing at all to do with incompleteness in the logical sense. The binary number system is perfectly capable of expressing all natural numbers. There is nothing surprising in the observation that in any base, there is a limit to what can be expressed with a single digit number. On the other hand, the incompleteness theorems were very surprising indeed. $\endgroup$ – John Coleman Mar 19 at 11:18
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    $\begingroup$ Having said that, there is a weak analogy in what you suggest, and analogies, even if weak, can potentially aid intuition, as long as you don't press the analogy too far. $\endgroup$ – John Coleman Mar 19 at 11:29
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    $\begingroup$ You are conflating objects and symbols. The number two is the object that has the symbol $2$ as name in the decimal system and the symbol $10$ as name in the binary system. $\endgroup$ – Mauro ALLEGRANZA Mar 19 at 13:01

Incompleteness (in the logical sense) is not about representation of mathematical objects. Rather, it concerns the relation of truth and provability in mathematics, where the latter concepts are understood in a specific technical sense.

There is no good metaphor which fully capture it. Douglas Hofstadter made an attempt in Gödel, Escher, Bach: An Eternal Golden Braid, which I recommend.

My advice is to begin by learning formal logic. Otherwise, it will be like trying to understand quantum physics without learning calculus or any other physics.

  • $\begingroup$ Thanks Daniel, now the 6th recommendation to me for that book and ironicially one that was brought up by my friend when I tried to think about this with him. $\endgroup$ – Rummy Mar 19 at 11:35
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    $\begingroup$ The book is very good. It was an inspiration for becoming a mathematician. $\endgroup$ – Daniel Ahlsén Mar 19 at 15:50
  • $\begingroup$ I have started on the Peano axioms/logic which will eventually go a long way. Unfortunately I know a few physicists already and do understand more about quantum physics than I should! I was formally educated in maths and physics until 18 though. Tbh I'm long term chasing maths that naturally crosses over (just for the enjoyment tbh) $\endgroup$ – Rummy Mar 28 at 7:41
  • $\begingroup$ Are you able to translate the incompleteness problem onto a more plaintext argument for a layman? Thats what im struggling with. I understand it to the extent of essentially that a formally rigid and complete system cannot be accurate because it has no room to deal with undefined terms, whereas an incomplete system can deal with and address unknowns due to iterative processes and thus it can be accurate, but it has to be incomplete because by definition because we cannot define the unknown until its...somewhat known? $\endgroup$ – Rummy Mar 28 at 7:52
  • $\begingroup$ Incompleteness is not about terms, knowledge or rigidity. It is about the possibility establishing the truth of arithmetical propositions via formal proofs. What it states is the following. If a formal system is such that (1) it's consistent, (2) it is strong enough to describe (a fragment of) Peano arithmetic, then (3) it cannot prove it's own consistency. $\endgroup$ – Daniel Ahlsén Mar 28 at 8:21

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