I have been wondering why exactly Hilbert asked for a decidable, complete, and consistent set of axioms for mathematics, rather than a decidable, complete, and sound set of axioms.

Consistency being: it is impossible to derive a contradiction from the axioms.

Sound being: the axioms are all true

Intuitively, it seems like you want the axioms to be sound, rather than merely consistent. For example, if we have a decidable, complete, and consistent set of axioms, but one of the axioms is that $\exists x \exists y \ x + y \not = y + x$, then that would not seem to be a very useful set of axioms about arithmetic. Or at least, it doesn't coincide with our intuition that addition is commutative ... we would intuitively say that that axiom is false.

So, I am trying to make sense why Hilbert was looking for a consistent set of axioms, rather than a sound set. Doing some research and thinking into this, I came up with several different explanations. I was wondering if someone could look at these and tell me if any of them actually make sense.

  1. The axioms having some property, and knowing that the axioms have some property are two different things. So, maybe Hilbert actually was looking for a sound set of axioms, but in terms of what we would be able to prove about the axioms, he was 'merely' looking for a consistency proof. In other words, Hilbert was looking for a decidable, complete, sound, and provably consistent set of axioms (and, I assume also provably complete and provably decidable)

  2. Hilbert was just much more concerned with consistency rather than soundness. Consistency is of course crucial: if the set is not consistent, you can derive any statement, and the whole foundation falls apart. Moreover, if you start with some 'obviously true' statements like the Peano axioms, then in order to keep the system consistent while we're adding more and more axioms, most likely the system will remain sound as well. So, as long as we make sure that things remain consistent, the set probably remains sound as well.

  3. Hilbert (and others at the time) were concerned about the very notion of 'mathematical truth': what is it, really? For example, things are true in Euclidian geometry that are not true in non-Euclidian geometry ... and it makes little sense to insist that one of them is 'more mathematically true' than the other. So, maybe we should say that the very axioms of a system define what is true in that system ... and that there is no notion of 'truth' beyond that; we don't step outside the system, and use some other measure of truth in order to declare: "Yes, those axioms are true", or "No, actually, that one axiom is not true". Instead, to Hilbert, the axiom set being 'sound' and 'consistent' was just one and the same. Indeed, whatever can be proven from a consistent set of axioms is thereby true, so likewise 'provable' and 'true' are the same as well. (I suspect there is a link with Logical Positivism here?) In an online paper "Hilbert's Programme", C. Smorynski states (top of p.7) that Hilbert said that:

If the arbitrarily given axioms do not contradict each other with all their consequences, then they are true and the things defined by the axioms exist. This is for me the criterion of truth and existence.

(Unfortunately, Smorynski does not give the exact reference for this ... it seems like it could come from Hilbert's Grundlagen Der Geometrie ... but some translation of course ... I would love to have an exact reference for this quote .. if it is indeed a quote)

So, which of these explanations is the correct one .. if any?

  • 3
    $\begingroup$ What do you mean by "true"? In Hilbert's program, there is NOTHING more fundamental than the "undefined terms", "definitions" and "axioms". What would you use to determine whether the axioms are "true" or not? $\endgroup$ – user247327 Jan 3 at 17:55
  • $\begingroup$ @user247327 Good question ... not sure how I would define true, but I would say 1+1=2 is true, and 1+1=3 is not true. But I was trying to figure out what Hilbert thought ... it seems like you prefer the second explanation then? $\endgroup$ – Bram28 Jan 3 at 17:57
  • $\begingroup$ 1+ 1= 2 in a particular number system but not necessarily in others. In Hilbert's system, and, in general, in abstract mathematics, a statement is "true" if it is one of the given definitions or an axiom or can be proven from the definitions and axioms. There is NO other criterion for "true". $\endgroup$ – user247327 Jan 3 at 18:07
  • $\begingroup$ Side point, but $1+1=3$ would cause an inconsistency in any reasonable system. False but consistent arithmetical statements are generally of the “there exists a natural number such that _” variety. $\endgroup$ – spaceisdarkgreen Jan 3 at 18:08
  • $\begingroup$ @spaceisdarkgreen Good point, I'll change my example ... $\endgroup$ – Bram28 Jan 3 at 18:13

A likely explanation is that this is for historical reasons:

Soundness is a notion that relates provability and semantics. A logical theory $\mathcal T$ is sound if whenever $\varphi$ is provable in the theory, then $\varphi$ is true in every $\mathcal T$-structure. This is a model-theoretic notion which could only be formulated once the basic notions of model theory had been developed – and this was only done by Kurt Gödel in the early 1930s as part of his work on completeness of first-order logic.

The notion of consistency is only concerned with provability, which is all Hilbert knew. A set of sentences $\Gamma$ is consistent, if $\Gamma \not\vdash \bot$, that is, we cannot prove falsehood from them.

  • $\begingroup$ Thanks! But maybe this just means that I misused 'sound' ... what I meant by a 'sound' axioms is that they are all true. So, to rephrase the question: why didn't Hilbert call for a decidable and complete set of axioms, all of which are true? $\endgroup$ – Bram28 Jan 3 at 21:39
  • $\begingroup$ The idea of probability and truth as different notions is later than Hilbert. $\endgroup$ – Hans Hüttel Jan 3 at 21:50
  • $\begingroup$ Hmm .. that would seem to be compatible with my explanation 3 ... that is, to Hilbert: as long as we don't have any inconsistency in the axioms, then whatever I derive from the axioms is thereby true. $\endgroup$ – Bram28 Jan 3 at 21:58

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