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Model theory is the branch of mathematical logic which deals with the connection between a formal language and its interpretations, or models.

60 votes
8 answers
6k views

Is the ultraproduct concept fundamentally category-theoretic?

Once again, I would like to take advantage of the large number of knowledgable category theorists on this site for a question I have about category-theoretic aspects of a fundamental logic concept. My …
Joel David Hamkins's user avatar
48 votes

What are some results in mathematics that have snappy proofs using model theory?

Plane geometry is decidable. That is, we have a computable algorithm that will tell us the truth or falsity of any geometrical statement in the cartesian plane. This is a consequence of Tarski's the …
Joel David Hamkins's user avatar
44 votes
2 answers
4k views

Is multiplication implicitly definable from successor?

A relation $R$ is implicitly definable in a structure $M$ if there is a formula $\varphi(\dot R)$ in the first-order language of $M$ expanded to include relation $R$, such that $M\models\varphi(\dot R …
Joel David Hamkins's user avatar
40 votes
Accepted

Has decidability got something to do with primes?

Goedel did indeed use the Chinese remainder theorem in his proof of the Incompleteness theorem. This was used in what you describe as the `boring' part of the proof, the arithmetization of syntax. Con …
Joel David Hamkins's user avatar
39 votes
3 answers
3k views

Can one show that the real field is not interpretable in the complex field without the axiom...

We all know that the complex field structure $\langle\mathbb{C},+,\cdot,0,1\rangle$ is interpretable in the real field $\langle\mathbb{R},+,\cdot,0,1\rangle$, by encoding $a+bi$ with the real-number p …
Joel David Hamkins's user avatar
37 votes
Accepted

Is it necessary that model of theory is a set?

You seem to believe that it is somehow contradictory to have a set model of ZFC inside another model of ZFC. But this belief is mistaken. As Gerald Edgar correctly points out, the Completeness Theor …
Joel David Hamkins's user avatar
36 votes
Accepted

In model theory, does compactness easily imply completeness?

There are indeed many proofs of the Compactness theorem. Leo Harrington once told me that he used a different method of proof every time he taught the introductory graduate logic course at UC Berkeley …
Joel David Hamkins's user avatar
32 votes
Accepted

Are the real numbers isomorphic to a nontrivial ultraproduct of fields?

The answer is no, because such ultrapowers are always $\aleph_1$-saturated, but $\mathbb{R}$ is not. More concretely, the ultraproduct will be an ordered field with uncountable cofinality — every cou …
Joel David Hamkins's user avatar
31 votes
Accepted

Is non-existence of the hyperreals consistent with ZF?

The answer is yes, provided ZF itself is consistent. The reason is that the existence of the hyperreals, in a context with the transfer principle, implies that there is a nonprincipal ultrafilter on $ …
Joel David Hamkins's user avatar
29 votes
Accepted

What are the advantages of the more abstract approaches to nonstandard analysis?

To my way of thinking, there are at least three distinct perspectives one can naturally take on when undertaking work in nonstandard analysis. In addition, each of these perspectives can be varied on …
Joel David Hamkins's user avatar
28 votes

Illustrating Edward Nelson's Worldview with Nonstandard Models of Arithmetic

If you are really satisfied with a model only of the theory $Q$, then you should be prepared for a bad situation, for this is an extremely weak theory. In fact one can make a computable model simply b …
Joel David Hamkins's user avatar
26 votes

Quantifier complexity of the definition of continuity of functions

It is truly a very nice question, one of those questions with an answer one feels must be right, but it is not so clear at first how to prove it. Nevertheless, aiming at partial progress, I claim that …
Joel David Hamkins's user avatar
26 votes
Accepted

Is multiplication implicitly definable from successor?

Contrary to my initial expectation, the answer is Yes. This answer is based on the idea of Clemens Grabmayer, which makes the observation that addition $+$ is definable from multiplication $\cdot$ and …
Joel David Hamkins's user avatar
25 votes
Accepted

Universal order type

You are looking for the concept of saturated model. A model $M$ is $\kappa$-saturated if any type consisting of fewer than $\kappa$ many assertions that is consistent with the elementary diagram of $M …
Joel David Hamkins's user avatar
25 votes

Non-definability of graph 3-colorability in first-order logic

Here is one way to do it. 2-colorability case. First let's warm up with the 2-colorability case. Notice that odd-length cycles are not 2-colorable, since the colors have to alternate as you go around …
Joel David Hamkins's user avatar

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