# Tag Info

### 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 ...
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### 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 ...
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### 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 ...

### 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 ...

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### 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 ...
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 $|{\cal L}|+\... 3 votes ### How to understand Tarski’s Real Closed Field theory result I'm not sure what you mean by "any real number combined with arithmetic operations and equalities or inequalities are decidable". Do you mean "the set of all real numbers combined with .... 3 votes Accepted ### Equational axiomatization and first-order axiomatization of the class of fields united with the class of Boolean algebras My first question is, is there a finite equational axiomatization of the variety generated by$K$, and if so, can someone give me an explicit finite basis? The answer to both of these questions is Yes,... 3 votes ### Equational axiomatization and first-order axiomatization of the class of fields united with the class of Boolean algebras Since both the first-order theories of fields and of Boolean algebras are finitely axiomatizable, so is the theory of the structures which are either fields or Boolean algebras. More generally, for ... 3 votes Accepted ### Theories coding finite sets Do you know some examples of theories that have FS, but do not come from Fields Theory context? Any theory which eliminates imaginaries has FS, so you can consider$T^{\mathrm{eq}}$for your favorite ... 3 votes Accepted ### Proving that every proper elementary extension of the real numbers has an infinitesimal element. Let$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 ... 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$\sigma(b)=b$... 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 ... 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:... 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 ... 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 ... 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 ... 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. ... 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$B$... 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 | ...
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 ...