How do we know that model existence implies consistency?

Question

For my honours dissertation, I have created an axiomatic set theory, which is a variant of ZF set theory, that has primitive ordered pairs as urelements (call it ZFP).

The axioms of the theory work nicely, and I have managed to construct a structure in ZF (domain, and definitions for the relation symbols) that satisfies the axioms of ZFP. The main point of doing this was to show that the set of sentences in ZFP is consistent, since a model can be proved to exist in an (assumed) consistent theory ZF.

However, I am struggling to find justification for this, and who these results are attributed to.

My loose argument is as follows:

Suppose we have a structure $W$, such that $W \vDash \varphi$ for each axiom $\varphi$ of $\mathit{ZFP}$.

$(1)$ Then for any sentence $\psi$, if $\mathit{ZFP} \vdash \psi$, then $W \vDash \psi$ (why?).

Suppose $\mathit{ZFP}$ is inconsistent, then for some $\psi$, $\mathit{ZFP} \vdash \psi$, and $\mathit{ZFP} \vdash \neg\psi$. But then by (1), $W \vDash \psi$ and $W \vDash \neg\psi$.

Is this then a contradiction by Tarski's truth schema, since $W\vDash \neg\psi$ is equivalent to $\neg(W\vDash \psi)$, which are both statements in $\mathit{ZF}$?

Am I at least on the right track? Or have I misunderstood certain definitions? Thank you.

Answer 1

Suppose we have a structure $W$ such that $W ⊨ φ$ for each axiom $φ$ of $\mathsf {ZFP}$.

Then for any sentence $ψ$, if $\mathsf {ZFP} ⊢ ψ$, then $W ⊨ ψ$ (why?)

This follows by soundness of FOL : if $\Gamma \vdash \psi$, then $\Gamma \vDash \psi$.

In your case, $\Gamma$ is the collection of axioms of $\mathsf {ZFP}$.

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Why model existence implies consistency?

The Model Existence Th asserts that a consistent theory has a model.

Let $\mathfrak A$ a structure and $\varphi$ a sentence (in general : a collection $\Gamma$ of sentences).

We define the satisfaction relation $\mathfrak A \vDash \varphi$ and we say "$\varphi$ is true in $\mathfrak A$".

We call $\mathfrak A$ a model of $\varphi$ (in general : a model of $\Gamma$).

We say that $\varphi$ is a logical consequence of $\Gamma$ (in symbols : $\Gamma \vDash \varphi$) when $\varphi$ holds in each model of $Γ$ (i.e. if $\mathfrak A \vDash \Gamma$, then $\mathfrak A \vDash \varphi$).

By the definition itself of the satisfaction relation : $\mathfrak A \vDash \lnot \varphi \text { iff } \mathfrak A \nvDash \varphi$, we have that in a model we cannot have that two contradictory sentences both hold.

This means that every theory having a model is consistent.