# Tag Info

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### 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 ...

### 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 ...

### Nonstandard infinite / hyperfinite sum in IST

You asked several questions that should really be separate posts, but let me start by answering one. You asked: "If anyone could provide a detailed proof that a sum indexed by an unlimited ...

### Is there a first-order sentence defining a cardinal?

Yes, here's one way to do it. It's kind of unsatisfying because it isn't succinct like the first-order characterization of an ordinal. As has been established, being an ordinal is a first-order ...

### If A is a substructure of B and A is isomorphic to B then A is an elementary substructure of B

The answer to your main question is no, and in fact $\mathbb{Z}$ itself gives us a counterexample: the substructure $\mathbb{E}$ of even integers is clearly isomorphic to $\mathbb{Z}$ itself, but is ...
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### When we say "cardinality of first order language L" and "cardinality of a structure or model" what we are meaning?

The cardinality of a structure means the cardinality of the underlying set of the structure. This is exactly what you should expect from ordinary mathematical usage: the cardinality of a group or a ...

### When we say "cardinality of first order language L" and "cardinality of a structure or model" what we are meaning?

The cardinality of the structure is the cardinality of the domain set of the structure. The cardinality of the language can mean one of two things: The cardinality of the set of non-logicial symbols ...

### Open source program that generate "random" theorems by exploration

I highly doubt you ever get any interesting theorem, even if you were to run such software for the remainder of your lifetime! Let's take something like Peano Arithmetic (PA). How do we enumerate its ...