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Thursday, October 14, 2010

Can the content of the theory be made relevant to its existence?

Can the content of the theory be made relevant to its existence?

If  so, we could have a hierarchy of theories based on how likely they are to exist. Such a hierarchy might be based on something like the following. Starting with the least likely to exist, consider T such that

(1) T ¢  Ø$T

I don’t know, hopefully there is no such T. If there were (and it was consistent) it would render the notion of existence within a theory irrelevant to the notion of actual existence. If there is no such T the notion of existence from within a theory is possible.

(2) T ¢  ($x)(x=x)

This T is the default degree of existence, because so many theories have this property. I don’t know if this is still valid when T is self-referential:

(3) T ¢  $T

This T would be still more likely to exist.

(4) T ¢  ƒ $T

This T would arguably be the most likely of all these theories to “exist”.

--Are there conditions stronger than (4)?

We want to rule out theories for which (2), but rule in theories for with (4). This is because we want to rule out the idea that all consistent mathematical structures exist. This would not be an explanation as to why any of them exist. Given that they all have the same existential status, why is it not the case that none of them exist?

One clue as to a way to proceed is that we want the why of its existence to be inextricably involved with its form (content). What does T have in (4) that it does not have in (2)? One thing is self-reference, but many sentences are self-referential. The salient property is condition (4), namely T stands in relation to the logical inevitability of its existence. Why should that matter? Because T’s existence is a necessary possibility.

(I have to remind myself: the reason we want condition (4) in the first place is that if we knew our universe was theory T, then it’s existence would be logically inevitable. The necessary existence of the universe would be a physical fact.)

What is a necessary possibility? Well, what is a possibility? Well, what does it mean to say such-and-such could be the case? In what ways does a possibility exist?

It seems plausible to say that a possibility exists the same way a number exists. Both of these are descriptions of reality.

Anyway, the first thing we do away with is possible-worlds semantics. Suppose we say í$T. It is not true that $T in some worlds and Ø$T in some disjoint set of worlds. What is true “in all possible worlds” is í$T. (Recall it is possible because we make no assumptions, so we do not assume nothingness is more natural than the existence of something.) Apparently

(5)  í$T  ®  ƒí$T

I don’t know if this respects the true-in-a-world and true-of-a-world distinction.

Then, the argument proceeds, the existence of the possibility of T is sufficient, in view of its peculiar definition, for it to bootstrap itself in to a valid notion of existence.

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