15
votes
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 ...
- 9,897
15
votes
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 ...
- 1,489
11
votes
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 ...
- 40k
8
votes
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 ...
- 40k
5
votes
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 ...
- 10.1k
5
votes
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 ...
- 240k
4
votes
Accepted
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 ...
- 73.7k
4
votes
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
...
- 56.7k
4
votes
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 ...
- 98.1k
3
votes
Accepted
Axiomatizing ZFC+Con(ZFC) without adding a constant symbol
Even though ZFC is not finitely axiomatizable, it is recursively axiomatizable: there is an algorithm to determine whether a given formula is an axiom of ZFC. Using this, we can define a predicate $\...
- 3,707
3
votes
Accepted
Is ZFC independent of the logic used?
Classical first order logic has many different equivalent presentations (not all of which are based on axioms!). One axiomatic variation can be found here, but really if you want to learn it you ...
- 56.7k
3
votes
Count the number of non-isomorphic models for a theory $T$ describing infinitely branching equivalence classes, each class infinite
I am still a little lost on how to intuitively understand the following definition of a type
A type (relative to $T$) is just a set of formulas which can be simultaneously satisfied by a tuple from a ...
- 73.7k
2
votes
Accepted
Prove that the set of nonprime numbers $>0$ is a spectrum
Here's an argument that generalizes nicely to similar sets in place of the composites, at the cost of resulting in less natural structures:
Take as our language $\{U,V,f\}$ where $U,V$ are unary ...
- 240k
2
votes
Accepted
Prove that divisibility between two natural numbers is not definable in arithmetic
We can do this by showing that the resulting theory is Peano Arithmetic and thus incomplete, while Presburger Arithmetic is complete.
We'll do this by showing that we can define multiplication if we ...
- 10.1k
2
votes
Accepted
Axiomatize the class of existentially closed structures
For inductive theories $T$, $T$ has a model companion if and only if the class of existentially closed models of $T$ is elementary (axiomatized by a first-order theory). An inductive theory satisfying ...
- 73.7k
1
vote
Accepted
$tp^\mathbb{Q}(a/\mathbb N) = tp^\mathbb{Q}(b/\mathbb N)$ iff there is an automorphism $\sigma$ of $\mathbb Q$ fixing $\mathbb N$ and $\sigma(a)=b$
Let $a,b\in\Bbb Q$ such that $\text{tp}^{\mathbb Q}(a/\mathbb N) = \text{tp}^{\mathbb Q}(b/\mathbb N)$.
Your proof that
$$\forall n\in\Bbb N\quad(n<a\iff n<b)$$
is correct,
and similarly
$$\...
- 29.4k
1
vote
Accepted
Proof that Skolem Arithmetic is a complete theory
As is the case with Presburger arithmetic, Skolem arithmetic satisfies a weak version of quantifier elimination (see e.g. Emil Jerabek's old MO answer), and this can be used to prove completeness. ...
- 240k
1
vote
Accepted
How to prove these three properties about sets?
You are correct about (b), it is likely a typo and they meant A × B ≠ ∅, instead. ( try proving (b) under the assumption that A × B ≠ ∅)
As for (a) your proof if fine, it relies on the fact that (p ⟺...
- 677
1
vote
Accepted
Enderton's "Mathematical Introduction to Logic": Is he proving second order induction?
When teaching "Intro to Proofs" if we want to go over induction on $\mathbb N$, we'll almost certainly opt to phrase everything in terms of full second-order induction, even though it's ...
- 56.7k
1
vote
Any formula is logically equivalent to a formula with all terms of height ≤ 1.
Long comment
The "height's def in this book may be a number of connective operations"; this is the height of a formula. Terms have no connectives.
For a term, the height (or depth) means in ...
- 93k
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