28
votes
How can different models of set theory be constructed from the same set of axioms?
It's helpful to think of the axioms of ZFC as being exactly like the axioms for groups. It just happens that the ZFC axioms are more complicated. Then, different "models" of the ZFC axioms ...
- 35.6k
9
votes
Accepted
Is "non-rigid" first-order axiomatisable?
Let $L' = L\cup \{\sigma\}$, where $\sigma$ is a unary function symbol not in $L$. Let $T'$ be $T$ together with axioms asserting that $\sigma$ is a non-trivial-automorphism (non-triviality is $\...
- 72.3k
9
votes
Accepted
Relations that ensure continuity
This answer was simplified by a lot from the original one.
Consider a relation defined as follows: $x \sim y$ if $x > y$, or $y = x + 2^{m}$ for some $m\in\mathbb{Z}$.
The idea is that the first ...
- 428
6
votes
Accepted
Regarding the "smallest" nontrivial, dense order type.
It's easy to see that $\eta$ embeds into any nontrivial dense linear order $J$ (of any cardinality whatsoever), at least assuming the axiom of choice: we can just use a "greedy algorithm" to ...
- 237k
6
votes
How can different models of set theory be constructed from the same set of axioms?
Then how can there be different models of the same set theory, based on the same set of axioms?
This turns out to be a very general feature of models of first-order theories, due to the compactness ...
- 409k
5
votes
Accepted
Does $\omega$-consistency depend on the encoding?
This is a great question! In general, "reasonably strong" theories will always have distinguished implementations (the jargon is actually interpretations, but meh) of the natural numbers, ...
- 237k
4
votes
Accepted
Must we define $\mathcal A \models (\varphi \wedge \psi)$ using the word "and"?
Defining satisfaction like this isn't wrong, per se -- in fact there's arguably no real distinction between this and the usual way -- but it puts undue emphasis on the formalization of the metatheory.
...
- 55.5k
4
votes
Accepted
Are the first-order theories of real addition (respectively, real multiplication) finitely axiomatizable?
I'll prove that $Th(\mathbb{R};+)$ is not finitely axiomatizable; a similar argument will work for $Th(\mathbb{R};*)$.
Let $\mathcal{U}$ be a nonprincipal ultrafilter on $\mathbb{N}$, let $p_n$ denote ...
- 237k
4
votes
Accepted
How to define an Ideal in the language of rings
It depends exactly what you want to do.
We can write a sentence in the language of rings together with a new unary predicate symbol $U$ such that whenever $\mathcal{R}$ is the expansion of a ring $\...
- 237k
3
votes
Accepted
Is it possible to prove the absolute consistency of a theory by giving a finite model?
I disagree with the premise that most consistency results are relative. I think there's a theory/meta-theory conflation going on here. Here are two standard examples of consistency results:
It's a ...
- 72.3k
3
votes
Accepted
Confusion about the model in Robert’s Nonstandard Analysis
The OP wrote: "Standard $\mathbb N$ doesn't contain any infinitely large integers." This is true, but $\mathbb N$ can contain unlimited integers. To clarify this, it is helpful to compare ...
- 38.2k
3
votes
How should one understand the "universe of sets"?
You may be interested in reading some of Nik Weaver's stuff, maybe Is set theory indispensable? and The concept of a set for starters. If I interpret his position correctly I think he agrees with you ...
- 409k
3
votes
Regarding the "smallest" nontrivial, dense order type.
There's a lot of overlap between my answer and Noah's, but I'm posting it anyway, because addresses your second question and contains some additional points about your first question.
A few comments ...
- 72.3k
3
votes
Accepted
Proof of Compactness Theorem’s failure in finite models
You write
"Because it talks about Finite Models that has a finite domain/universe, therefore the number of constants have to be finite"
This isn't true: there's no rule that says that you ...
- 237k
3
votes
Bound for the cardinality of the model of a set of formulas
What if there is no infinite model $M$ of $Σ$, Only finites? For example let $Σ=\{∀x\forall y (x=y)\}$ has only models of cardinality $1$, certainly it has no models of cardinality $|{\cal L}|+\...
- 9,948
3
votes
How to understand Tarski’s Real Closed Field theory result
I'm not sure what you mean by "any real number combined with arithmetic operations and equalities or inequalities are decidable". Do you mean "the set of all real numbers combined with ....
- 12.4k
3
votes
Accepted
Equational axiomatization and first-order axiomatization of the class of fields united with the class of Boolean algebras
My first question is, is there a finite equational axiomatization of the variety generated by $K$, and if so, can someone give me an explicit finite basis?
The answer to both of these questions is Yes,...
- 13.2k
3
votes
Equational axiomatization and first-order axiomatization of the class of fields united with the class of Boolean algebras
Since both the first-order theories of fields and of Boolean algebras are finitely axiomatizable, so is the theory of the structures which are either fields or Boolean algebras. More generally, for ...
- 237k
3
votes
Accepted
Theories coding finite sets
Do you know some examples of theories that have FS, but do not come from Fields Theory context?
Any theory which eliminates imaginaries has FS, so you can consider $T^{\mathrm{eq}}$ for your favorite ...
- 72.3k
3
votes
Accepted
Proving that every proper elementary extension of the real numbers has an infinitesimal element.
Let $R$ be a proper elementary extension of $\mathbb{R}$, and $r\in R\setminus \mathbb{R}$.
Case 1: $r>q$ for all $q\in \mathbb{Q}$. Then $1/r$ is a positive infinitesimal.
Case 2: $r<q$ for ...
- 72.3k
2
votes
definability and algebraicity over canonical base
It seems that this is just stationarity (if the following is correct?):
Let $\varphi(x,b,c)$ be witnessing that $a\in\text{dcl}(C,b)$.
Suppose that $\sigma$ is an automorphism such that $\sigma(b)=b$ ...
- 51
2
votes
Is the Null Set Axiom necessary in ZF?
If you are working with a form of first-order logic where $\vdash \exists x (x = x)$, and if you moreover postulate the axiom scheme of separation, then you are correct that the null set axiom is ...
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2
votes
How to understand Tarski’s Real Closed Field theory result
I think there's some confusion here about what decidability of $\mathsf{RCF}$ means.
The theory of real closed fields, $\mathsf{RCF}$, is a theory in first-order logic in the language of ordered rings:...
- 72.3k
2
votes
Quantifier elimination in ACF from an example
I'm not sure this is a complete answer to your question (in particular, I don't work out any examples), but I have a number of comments to make, which I hope will be helpful. Let me first point out ...
- 72.3k
2
votes
How to define an Ideal in the language of rings
I wanted to formalize the notion that if I have some property expressed as a sentence $\phi$ that is true for all ideals in a commutative ring (hence the need to quantify over ideals) such that the ...
- 409k
2
votes
Is "non-rigid" first-order axiomatisable?
Yes, you can add a new unary function symbol $g$ and then write down a schema that says "$g$ is a nontrivial automorphism" and add that to the schema saying "$M$ is infinite" and ...
- 55.5k
2
votes
How can different models of set theory be constructed from the same set of axioms?
The theory of mammals includes several axioms:
Female mammals produce milk
Mammals are warm-blooded
Mammals have hair
There are many creatures that satisfy these axioms and are therefore mammals. ...
- 3,083
2
votes
Accepted
Elementarily equivalent expansions of structures
Your last paragraph is moving in the right direction, but is mixing up elements and constant symbols: when we write $$(A,\overline{a})\equiv(B,\overline{b})$$ we mean that the expansions of $A$ and $B$...
- 237k
2
votes
Quantifier elimination in ACF from an example
For question 1 in the general case ($P$ and $Q$ may have coefficients involving other variables), we're looking to determine if $P$ has a root that $Q$ does not have. This is equivalent to $\neg (P | ...
- 2,646
2
votes
Quantifier elimination in ACF from an example
Let me give a simple example of what one step of a quantifier elimination algorithm could look like. The basic idea is that in a generic case, an ideal of $\mathbb{F}[x_1, \ldots, x_{n-1}, x_n]$ will ...
- 19.6k
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