Where do models of false theories exist?

Question

I have some difficulties in understanding the [uni]verse platonic view. How are we to understand the existence of a model of a false theory? what is the relationship of this model to the platonic world of sets?

My personal try is the following, let's assume the existence of a platonic universe $P^{sets}$ of sets, and also another universe $P^\in$ that serves as a relational universe, i.e., that can stand as a representative of the relation epsilon $\in$ in the real world. For example $P^\in$ might be a universe of some object representation of "ordered pairs" of sets, these ordered pairs are primitive ones, they stand for really existing objects exemplifying ordered pairs in a platonic realm. So $P^\in$ would be a really existing platonic membership relation between objects of $P^{sets}$.

Now when we say for example the rules of $ZFC$ are true, in a platonic sense, would that be taken to mean that the sets in $P^{sets}$ would be related to each other by the platonic membership relation in a manner that abides $ZFC$ rules.

So now when we say that $ZFC$ can interpret $ZF\neg C$ then this mean that the later is consistent (should $ZFC$ be consistent) so there would exist a model of $ZF\neg C$, but how I'm to understand the existence of this model in the platonic world? is it the case that the interpretation would be a pair of subsets of $P^{sets}$ one representing a domain and the other is a set of set-ordered pairs (non primitive ones, i.e. interpreted ordered pairs as special kinds of sets), and so the rerlation object interpreting the membership relation is not a real relational one, i.e. not composed of primitive membership ordered pairs??? as it is the case with ZFC? so this model is deemed as not real?

Is that a good analogy? I mean I want to conceal two matters: that there EXISTs a model of a false but consistent theory, if we are to hold platonistic views, then this existence must be in some reality? otherwise the assertion "there exists" wold have no meaning, now this reality must be taking part in the true platonic world of sets? where else it would be? On the other hand we must differentiate this kind of existence of a model of a false theory from the existence of a model of a true theory! that's why I used the "primitive ordered pairs" to represent the relations in the theory, a true theory would have its relation sets being real in the platonic sense, i.e. composed of primitive ordered pairs abiding the rules of that theory, while a false one would not have them real, i.e. the relation on its domain is not composed of primitive ordered pairs.

I always had the following impression:

for every effectively generated theory $T$ we have: $$ Con(T) \to \exists T^* (True(T^*) \wedge T^* \ inteprets \ T)$$

the idea can be seen in the above anaology, if a set theory $T$ is consistent but false, still the model of $T$ must be a part of $P^{sets}$ and so there must be a true theory $T^*$ (i.e. $T^*$ satisfied in $P^{sets}$) that can itnerpret $T$, otherwise how can we explain that a model of $T$ must exist? exist where?

Is that impression correct?

I visualize the whole platonic world of mathematics $P^{math}$ as the union of the platonic worlds of the individual disciplines i.e. the union of $P^{arith}$ , $P^{Geom}$ , etc..., I think standard models of those disciplines are the real ones represented by the Platonic realms of those disciplines, i.e. they have real relation sets, as opposed to the fake or false theories which don't have real grounded relations in the platonic world.

I know that this question is in some sense philosophical, but "platonism" especially the [uni]-versed one, is an almost ongoing working assumption of most innovative work in mathematics and set theory, so it is important to at least shed some light on that aspect.

Answer 1

I'll answer this from a Platonic viewpoint. Consider a theory $T$ that is false when its primitive concepts are interpreted in the standard way. For example, $T$ could be the theory ZF$\neg$C that you mentioned in this connection. If we interpret the primitive concepts "set" and "element" in the standard way, so that "set" refers to all the sets in our Platonic world and "element" refers to the actual elementhood relation between these sets, then the axiom of choice is true, so ZF$\neg$C is a false theory. But, because it is consistent, there are other interpretations of the primitive concepts that make ZF$\neg$C true. That is, one can define (within the Platonic universe, using the standard notions of set and element) concepts that I'll (for the purpose of this answer) call "pseudo-set" and "pseudo-element" inn such a way that all of the axioms of ZF$\neg$C become true when one interprets "set" and "element" to mean "pseudo-set" and "pseudo-element". More formally, if you take any axiom of ZF$\neg$C, replace every $\in$ in it with the "pseudo-element" relation, and replace every quantifier "for all $x$" or "there exists $x$" with "for all pseudo-sets $x$" or "there exists a pseudo-set $x$" (respectively), then the resulting statement is true in the Platonic universe (and in fact is provable in ZFC). So the answer to "where do models of false theories exist" is "in the Platonic universe" --- but they depend on using unusual interpretations of the primitive terms of those false theories. To put it another way: A false theory becomes true if its primitive concepts are suitably reinterpreted. (A Platonist might well say "misinterpreted" instead of "reinterpreted".)