All Questions
Tagged with model-theory axioms
58
questions
3
votes
1
answer
396
views
Does Löwenheim-Skolem require Foundation in any way?
As title states, I'm curious whether Löwenheim-Skolem (in either of its upward or downward versions) necessitates some implicit use of Foundation. The usual presentation makes quite clear the reliance ...
- 512
4
votes
1
answer
137
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How can we verify if a structure statisfies ZFC or not?
I'm studying set theory which leads me to some confusions of model theory.
For the context, I know that the set $\mathbb{N}$ with the usual operations $(0,+,×)$ is a model of Peano arithmetic. It ...
- 1,350
1
vote
0
answers
64
views
Connection of axioms of first order logic and axioms of first order theory
If we have a set of sentences S in first order logic. We know that we can create a first order theory Th(S) from S, which is the "set S" union "the sentences which we can prove them ...
- 11
5
votes
3
answers
1k
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How can different models of set theory be constructed from the same set of axioms?
A beginner question here, sorry if it seems obvious.
From my understanding, a set theory like ZFC strictly dictates what can and cannot exist, e.g. the Axiom of Infinity implies the existence of an ...
0
votes
2
answers
155
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Can every set in the Von-Neumann universe be obtained from finite ZFC steps [closed]
For me, ZFC feels like saying either something is a set, or from a set, we know another thing is a set. On the other hand, something like doing the powerset operations for $\omega$ a number of times ...
- 1,722
4
votes
2
answers
499
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Functions that cannot be written as functional formulas and the axiom of replacement
A formula $\phi(x, y)$ defined a function on a set X if for each $x \in X$, there is exactly one set y such that $\phi(x,y)$ holds. We then say $\phi$ is a functional formula, in the sense that it ...
- 1,722
3
votes
1
answer
170
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Is there a tangible countable model of set theory
ZFC or NBG set theory have a countable model by Löwenheim-Skolem. But I never found an example of one with real tangible sets. Of course I do not look for a consistency proof.
I am seeking to see how ...
0
votes
1
answer
41
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Semantic Interpretation of the axiom of restricted separation
In a follow up to this answer to a previous question, I want to see if I have a correct understanding now of the semantic interpretation of the axiom of restricted separation. As a side note I'm also ...
- 855
0
votes
1
answer
88
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Power sets vs definable power sets in the minimal standard model of ZFC
As noted in this answer by Eric Wofsey, as well as the Wikipedia page on the Constructible Universe, we have that $L_\alpha$ is strictly smaller than $V_\alpha$ for any $\alpha > \omega$ unless $\...
- 8,000
1
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1
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73
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Stronger versions of the Power Set Axiom?
In this article on Skolem's Paradox, it gives the Power Set Axiom as an example where models may badly misinterpret the axioms (compared to the "intention" of the axiom). While originally I ...
- 1,540
1
vote
2
answers
180
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Is the theory for $\mathbb{R}$ categorical or not?
The usual axioms of $\mathbb{R}$ consist of the ordered field axioms plus the least upperbound property. Because the least upperbound property is a statement that quantifies over sets of real numbers, ...
- 2,461
2
votes
1
answer
504
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Concrete and abstract models of axiomatic systems
In order to prove the consistency of an axiomatic system we must come up with a model. Wikipedia gives the following definition for a model of an axiomatic system:
A model for an axiomatic system is ...
- 256
11
votes
2
answers
655
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What exactly is the distinction between a theory and model in model theory if models are themselves constructed in axiomatic theories?
My understanding is that model theory requires a distinction between a logical theory and a structure to interpret the statements of the theory.
However, every piece of mathematics including every ...
- 2,461
3
votes
1
answer
145
views
How do model theorists define structures?
Let $L$ be a first order language and $A$ be an L-structure. Let $\sigma$ be an L-sentence.
In most of the mathematics I have seen a structure is defined by the axioms it satisfies. These may or may ...
- 105
1
vote
1
answer
44
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Non-contradictory axiom system for a binary operation
Suppose we want to define a binary operation $\otimes:\mathbb{N} \times\mathbb{N} \rightarrow \mathbb{N}$ on a ring $(\mathbb{N},+,\cdot)$ with an arbitrary system of axioms. The axioms may be given ...
- 503