# Some questions about implications of Gödel's Completeness Theorem [closed]

I have read Gödel's original paper (1930 - reprinted into J.van Heijenoort (editor), From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, 1967) but I'm still unsatisfied with my understanding of it.

Modern logic textbooks usually prove the theorem following Henkin's construction.

One of the peculiarity of Henkin's proof is that the "completeness" aspect (i.e.if A is valid it is provable) is somewhat of a by-product of model existence.

Gödel's proof use (natural) numbers. This is obvious (with insight) today that we know about Gödel's philosophical realism.

Hankin's construction avoid numbers but use the "syntactical stuff" to build the model. But this (according to my understanding) is not really different; in order to "run" the construction we need countable many symbols, and symbols are "abstract entities" (like numbers). I think that we really needs them : we cannot replace them with "physical" tally marks. So my question :

In what sense we can minimize the "ontological" import of the theorem ?

In a previous effort I asked for some clarifications to a distinguished scholar and I received this answer : "About the completeness proof, it is a theorem of orthodox mathematics, and does not pretend to be nominalistic."

• Can you frame the headline here as a question? I tried to and sort of got stuck, there's a lot going on here (it's maybe something about "minimizing ontological implications" of incompleteness?) -- I'm marking "unclear" for now Jan 17, 2016 at 22:29
• You may want to ask some questions about this on the mathematical form. As for the "syntactical stuff" argument, there are very specific requirements to invoke Godel's incompleteness theorem. The main one is omega-consistency, which is, in layman's terms, "the system can prove all true statements in the arithmetic of natural numbers." If it can do that, then it really doesn't matter what sort of clever tricks were done to obscure it (unless you do something really clever like Dan Willard's tricks to prohibit the construction of diagonalization) Jan 18, 2016 at 2:41
• @MauroAllegranza Some philosophers/mathematicians argue that Gödel's Completeness Theorem has limited relevance to questions of broader metaphysical significance. While Gödel's theorem has deep implications for the study of mathematical truth and logic, its implications for questions about the nature of reality or existence are indirect and subject to interpretation. As such, minimizing the ontological import of Gödel's Completeness Theorem involves recognizing its specific domain of applicability and its limitations in addressing broader metaphysical questions. - says ChatGPT in summation. Mar 4 at 9:02
• It sounds legit to me, but I am not a student of the history of philosophy. (didn't much care for it). More into the practical. Ao you will have to be the judge of whether ChatGPT came up with the right suggestion... awareness of domain of applicability, Mar 4 at 9:03

## 1 Answer

There's a sense in which, if you want to do away with the ontology then the completeness theorem (along with pretty much the rest of mathematics) would be strictly speaking false. Expressing the theorem in terms of validity:

If T |= φ then T |- φ

is trivially equivalent to expressing it in terms of models:

If T is consistent then T has a model

But, if you believe that there are not infinitely many objects (in the form of numbers, abstract syntactic objects, or whatever) -- which you may well do if you are a nominalist -- then you should believe that this is false. For there will be theories (such as Peano Arithmetic), which are consistent, but which can not have a model (since there aren't enough objects to go around).

• Good point. The issue I'm thinking of is something like this: (i) from a "foundational" perspective, the metamathematical program of Hilbert was well-concived (but defeated by Godel's Incompl.Th) : the original program was aimed at "bootstrapping" the mathematical building from finitist ground; if it is impossible to prosecute along this way, (iia) or we stay with a finitist approach, or (iib) otherwise we need on ontology of mathematical entities based on at least (natural) numbers, assuming at least the (potential) infinity of counting. Regarding the (actual) infinity of the real line... Jan 11, 2014 at 10:41