Skolem's paradox

So far I never mentioned how I understand
the famous Löwenheim-Skolem result (*):

"no matter how fancy your axiomatic system, which seems to talk about real numbers, complex numbers, geometries, fields etc.,

in the end, all it really does is talk about the countable natural numbers, nothing more and nothing less."

In my opinion it is the most shocking result of the Grundlagenstreit.

But does this tell us something about the true nature of physical reality?

(*) and let me be very clear that I am a layman who read exactly one book about number theory (and I understood perhaps half of it).


HereticRex said...

I hope you're still watching your blog.

Couldn't we find an understanding this way?

No matter what semantic valuations the terms of the paradox's language have, if the syntax of the language is countable, then it must necessarily be "down-skolem-izable" to a countable semantic domain.

And: So far, all of our languages are countable, in this sense--our alphabets are all finite, and our sentences are all finitely long strings of our alphabets. Furthermore: At any given time in history, the number of occurrences of each of our many (but still finite) languages, is necessarily finite. So every occurrence of a language which interprets an uncountable domain has a syntax, and a history, which is satisfied by a countable domain.

And: Isn't it generally accepted that the semantic domains of a language may be chosen arbitrarily, and still yield to up- or down-skolem-izing? That is, isn't skolemizing thought to be something like universally sound?

(I don't want to sound like a finitist here--I depart from them, in the view that notwithstanding the fact that we have no syntactically uncountable languages, such languages are possible, hopefully for humans, but nearly certainly for enlightened aliens with a billion years of technology and meditation on logic, if there ever was or will be such a civilization.)

Of course, fixed points might mix this work up, seeming as they do, to "fix" (in the sense of "gluetogether" or "equate") the syntax to the semantics. This much appeared right away, in Goedel's "On undecidable propositions..."

My teachers (Woods, Brown, Dunn, Barwise, Feferman, ...) were downright evasive about this line of thought. (Except I leap to note that Woods seemed to want to put the question on to the others rather than starve it to death.)

James D.K.

HereticRex said...

(This note is just an attempt to adjust my settings at your blog to email me when there is a new comment.)

wolfgang said...

>> all of our languages are countable

and it is not surprising that every axiomatic system of set theory (and its statements) is formulated in such a language.

what is surprising (at least to me) is the fact that every such axiomatic system actually has a countable model.

HereticRex said...

(Part 1/2)

After I understood and realized the fact that some infinities are larger than others, it surprised me, rather, that logicians believe that finite axiomatic systems can have uncountable models.

It's been a long time since I've done this kind of work, but it still occupies a lot of my thinking. To my understanding, Skolemization depends, de facto, on the so-called diagonal method. Diagonal argument is always (again, to my understanding) a potentially enumerative argument, establishing that one infinite set is smaller (or larger) than some other infinite set.

Take Goedel for example. His proof can be understood to show that the Goedel sentence is not among the set of the well-formed formulas of the Principia Mathematica. The problem, here, for logic and logicians, is that the Goedel sentence certainly does take the form of a well-formed formula. Actually, it can be taken even to demonstrate an even more stark falsehood: that the Goedel sentence is not even a string of the alphabet of the Principia.

So it is not at all surprising that logicians, then and now, have opted to "solve" the conundrum by declaring it to be formally "undecidable", and thus, a dark smelly matter that must be declaimed in all cases and circumstances, and behave as if logic has no such flaw as a well-formed analytic and provable falsehood. (It remains a flaw only for any who may mistake it for a decidable proposition, but those people are branded by a tint of Orwellian doublespeak, as if full-blown heretics.) So it is accepted, against human reason, that not all analytic truths are provable.

HereticRex said...

(Part 2/2)

This is a sad matter for me. It seems to me that this is the victory of tenured unreasoning hubris over human understanding. I met an astronomer once who is keenly interested in my take on these matters, but who seems to be prevented by professional prudence to enter the argument. I would dearly like to know how he would enter the debate, what difference it makes to physics and astronomy.

It seems to me that there is a very economical solution to the problems I would have raised in the professional community. We might simply say that formal logic is fine as it is, consistent and complete, but except only that it does not remain consistent for uncountable semantic domains. I can only suppose that uncountable models are very dear to the inheritors of formal logic, such as physicists, or maybe others, and that this blocks simple solutions, not for the sake of truth, but for something even more dear, whatever that may be. It has been pointed out to me time and again that the scientific method proves the utility of formal logic just as it is, undecidable propositions and all--logic has no place, they argue, teaching a scientist anything. "If it ain't broke, don't fix it."

But it is broke.

This is pretty much the limit of my technical skill--I could address all the niggly details if I had the time, and mixed with the professional community, and thus had sincere commentary from them, but that just isn't so. My comments have already been met by intellectual malice and violence.

If the cat is on the mat, but I'm not allowed to say so, then there's no air for me to breathe.