[FOM] On Existence of Mathematical Objects

Vladimir Sazonov V.Sazonov at csc.liv.ac.uk
Thu Oct 9 15:42:19 EDT 2003

Dmytro Taranovsky wrote:
> In my last posting, I demonstrated that it is generally agreed that we
> can refer to mathematical objects, and I noted that this implies that
> mathematical objects metaphysically exist.  Vladimir Sazonov objected,
> claiming that mathematical theories are like works of fiction, where we
> can talk about the characters without claiming that they exist. 

More precisely, if you will take my posting as the whole, you will 
see that I even agree that all these fictions, as well as electrons 
and mathematical objects has some kind of existence, but quite 
different. When we say "exists" we should be sufficiently precise, 
in which sense. Many of them exists only in your, my and any other 
person's imagination and have no immediate relation to the real 

> However, "object exists" means "object is" which means "object is
> something", so "object does not exist" means "object is not" which means
> "object is not something" which means "object is nothing".  If you refer
> to an object, you do not refer to nothing, so you claim or assume that
> the object exists.  

Sorry, I consider this rather as some linguistic exercises. 

On the other hand, the fact that the set of all real
> numbers exists does not mean that it has corporeal existence--it
> certainly does not;  and although, technically speaking, the fact that
> we can meaningfully refer to a character in a work of fiction implies
> that the character exists, a fictitious character cannot cause anything,
> so, as a practical matter, we can assume its nonexistence.

Of course, fictions have some influence on our minds. 

> An objection to Platonism is the claim that there are multiple universes
> of sets and that,  for example, the Continuum Hypothesis is meaningless
> because it is true in some universes and false in others, and in talking
> about the Continuum Hypothesis, we fail to specify which universe we are
> talking about. The objection appears reasonable--after all, sentences
> about fictional characters are often meaningless because the work of
> fiction examined only partially specifies the fictional universe that
> the work is about and various universes consistent with the work of
> fiction may contradict each other

The same as in mathematics, except in mathematics these fictions 
behave according to some explicitly specified formal rules. 
Thus, I do not agree with the following:

--except for the fact that in talking
> about sets, we refer to the unique maximal universe that consists of all
> sets that exist.  Unlike a work of fiction, where parts of the plot may
> be unspecified, the invocation of maximality and totality causes the
> universe of sets to be unique.  For example, suppose that the notion of
> the set of all real numbers is vague.  In that case, there are two
> different sets, R1 and R2, each of which is the set of all real
> numbers.  By the axiom of extensionality, there is a real number r that
> belongs to only one of the two sets.  However, the set that does not
> contain r cannot be the set of all real numbers, contradicting the
> assumption.

How it is possible at all to apply the axiom of extensionality 
in this context? 

> The issue of multiple universes of sets does explain the fact that so
> many important questions are undecidable in ZFC.  

Just vice versa, it explains! Axioms cannot (completely) fix 
in general what are we talking about. 

Figuratively speaking,
> the deductive apparatus of ZFC does not know that ZFC is meant to be the
> theory of all sets 

ZFC "does not know". But who knows? You? Which way? 
Is it just your belief or a knowledge? Even if it is 
a belief of an overwhelming majority of people, why other 
people should follow them? Why should we use ANY beliefs 
in science at all? I do not say here about imagination. 
This is quite different thing. We can imagine anything 
without any belief in what we imagined. All mathematics 
can be understood and practically developed in this way. 

and considers a statement to be a theorem only if it
> is true in every universe 

It is difficult and even impossible to speak of "every universe" 
in this context. I and you have an imaginary universe of ZFC 
(without which it would be impossible to do anything in ZFC). 
But I have no idea how to compare your and my universes. I am even 
unsure that I myself have any fixed universe, just some idea, no 
more. The great Cantor also had some very nice idea. It proved to be 
contradictory. But it is still very nice. However, it seems, 
everybody will agree now that it is a vague idea. I do not see 
any essential difference between this idea and the idea of natural 
numbers. May be PA will be once shown to be contradictory, too. 

In your other posting you wrote: 

> An inconsistency in PA arithmetic would be a disaster--

It would be disaster only for those who have superbeliefs. 
Just as it was in the case with the old paradoxes in set theory. 

most fields of
> knowledge and practical constructions depend on arithmetic (although
> most practical applications can actually be carried out in weaker
> systems).  

Yes, of course. This or other way the most important mathematical 
considerations will survive. We will probably get a very good 
lesson on the general philosophy of mathematics. 

Fortunately, we know a priori that every axiom of PA is true,
> and hence PA is consistent.


Let us continue with your current posting

that satisfies the basic closure conditions
> enumerated in ZFC.  The solution is to find new set existence axioms and
> closure properties:  Since the universe consists of all sets, it should
> satisfy strong closure properties,  and since every set that exists in
> some universe actually exists, 


it should also satisfy strong existence
> axioms.  The universe, being unique 


rather than random, should also be
> canonical.  


The difficulty is that we can know which statements are
> subtle approximations to the claim that every set exists or that the
> structure of the universe is canonical only after we carefully research
> their consequences and their subtle relationship with other
> propositions.  

I think, I should not spent time to substitute the above 
question marks by some self-evident replies. 

I would suggest you, instead, to make something concrete from 
your considerations. Let S be any reasonable extension of ZFC 
where it is provable existence of a model of ZFC. Of course, 
S is stronger than ZFC. Let us consider all models of ZF 
in S. Of course, from your point of view these are not ALL 
models of ZF. Strictly speaking I have no idea what does 
it mean this "ALL". But in the framework of S we can consider 
quite rigorously all models of ZF, like all (not ALL) ordinals, 
cardinals, etc. I hope you understand the difference I make 
between all an ALL here. 

Of course, some of this models of ZF may have as an element 
a proof of a contradiction in ZF or even in PA. This follows 
from Goedel theorem. Moreover, some of these models M of ZF 
may have their (internal) \omega_M which, from the point of 
view of S are not isomorphic to the \omega of S. Let us exclude 
them from our considerations. (I think this would be also your 
own choice.) Thus, let us consider in S all models M of ZF with 
the standard \omega_M. May be some further restrictions should be 
imposed on these models of ZF (in S)? I do not know. May be, 
it will be simpler to consider only models of ZFC? 

For some of these models AC (or CH) will hold and for other 
will not, etc. 

Now, the proper question to Dmytro Taranovsky (and anybody else). 

Is it possible to unify all these models M of ZF (in S) in one big 
model (let it be even a class, not a set like each of these M) in any 
reasonable sense so that we would be able to say that we have got a 
model of ZF consisting of all (I do not mean ALL) sets. In particular, 
we could probably get this way a "union" (it is evidently 
meaningless to use the normal concept of set union in this context) 
of all real lines R_M internally and canonically existing in each M? 

Of course, this will not give us the universe of "ALL" sets 
or the "set" of "ALL" real numbers as Dmytro Taranovsky 
wants to have. But, anyway, is it possible to do what I suggest? 
May be some partial interesting solution exist? 

Dmytro, if you are interested in this exercise (which seems 
is in a coherence with your views) and will succeed, will you 
tell us what you will get? May be some negative result will 
be obtained? It would be also interesting. 

If you are interested, good luck! 

Vladimir Sazonov

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