# Tag Info

Accepted

### Are there more truths than proofs?

That's correct, but it is not very interesting: we don't have "access" to most mathematical facts. It is not even clear what a proof of an arbitrary mathematical fact would mean, because to ...
Accepted

### How do we compare the set of natural numbers in different models of ZFC

We cannot compare the $\omega$s in different models, can we? If we say that one is embeddable into another, an embedding $\omega\rightarrow\omega'$ between the two sets of natural numbers in different ...
Accepted

### Nonstandard infinite / hyperfinite sum in IST

Welcome to Math.SE! You did not indicate how much background you have in Internal Set Theory (IST) - but based on what you wrote, it seem like you have a few misconceptions about it. Let us start by ...

### Understanding ZFC

ZFC does generate a “theory”. Formally speaking, a theory is simply a set of sentences in the given formal language. Any set of axioms generates a theory, namely the collection of sentences provable ...

### Are there more truths than proofs?

The argument is fallacious. We can use the same basic argument in an even simpler setup, which clearly indicates what goes wrong. There are uncountably many subsets of the natural numbers. There are ...

### Is proof of the law of identity a case of circular reasoning?

I don't understand where you think the circularity is coming from. We have an intuition about equality, namely "everything is equal to itself". We'd like to formalize this intuition as a ...

### Don't Understand Mechanism behind Proof by Contradiction

For the second case, the argument runs by cases. Roughly, it is: P1. $A\lor \neg A$, by the principle of the excluded middle. P2. $A \implies \neg A$, by assumption. By applying P2 to P1, we have: L3. ...
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### Is FOL really sound?

First let's be clear about the fact that there are two levels at play here. At the top level (sometimes called the "meta-level"), there is the logical reasoning that we human mathematicians ...

### Understanding ZFC

You asked: If one is to do mathematics rigorously, they would like every concept they mention, such as a function or relation, to have a precise definition. How can we do that with ZFC if ZFC does ...
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### Is there a technical reason to require relation symbols to have positive arity?

Purely historical/conventional. There is no technical reason to exclude $0$-ary relation symbols (proposition symbols). I can think of two reasons why model theorists tend to ignore proposition ...