Re: An uncountable countable set

Aatu Koskensilta wrote:
Tony Orlow wrote:
Well, I understand that, but it doesn't seem to me that one CAN measure an infinite set with any accuracy without involving some notion of measure into the properties of the elements which define the set.

One obviously can, in so far as cardinality can be said to be a "measure" of an infinite set. Whether comparison of sets in terms of cardinality meets the criteria associated with some notion of "measurement" can of course be questioned, but has pretty much nothing to do with modern set theory.
Can cardinality be said to be "accurate", in the sense that it detects all changes in set size? Not if a proper superset is not larger. It's a gross measure, a classification scheme, but not an accurate measure of any sort.


Cardinality gives a certain gross measure of complexity, but to consider the alephs to be any exact numbers of any sort is unjustified, in my opinion.

You can consider them inexact numbers if you want. This has no effect on the mathematical content of the set theoretical theory of cardinality.

But it does have an effect on how cardinality is treated. If set theory is a generalization of all mathematics, then none of the general conclusions it draws should be at odds with any of the specific conclusions drawn throughout mathematics. That is, if it applies to all mathematics, then it applies to a superset of any field of mathematics, and should not contradict any such subset.


Yes, there are some concepts such as limiting density and Lebesgue measure, which come to different conclusions regarding the same sets. That is, they detect differences between some sets where cardinality does not. Doesn't this mean that bijection alone misses real distinctions in element count, or size, which can only be detected using more sophisticated methods?

Cardinality is one property of sets, and obviously there are others - which might in some context be more appropriate measures of "size" for some mathematical purposes. But as said, when we consider infinite sets notions such as "size" are ambiguous, and we might well get different answers for different mathematical explications of different informal ideas of what "size" means, even when those ideas are equivalent when applied to finite sets. In case of general abstract set theory in which we're dealing with sets that do not come with associated structure cardinality is the only notion that can be universally applied.

Can you give an example of transfinite set theory determining the cardinality of an infinite set with NO reference to the stricture of the set? Even the most basic limit ordinal is based on an inductive structure based on the successor function.

If you're not interested in that, you're of course free to study structured sets and devise measures that in some way better reflect some informal notions of size associated with those sets and structures. If your pursuits in this directions are in some interesting way connected to something mathematicians find interesting, it is possible that they would take interest in your new notions of size, but if you just go on about how cardinality misses something without connecting your criticism to actual mathematics, or produce interesting new mathematics - possibly in some alternative framework - you'll just be ignored by everyone but people like Virgil, who are perhaps even more eccentric than you. (You might have noted that I haven't had anything substantial to say about your ideas, and have only offered general reflections, for exactly that reason.)

Yes, you and I have not talked a lot in the past. I don't think you are familiar with my IFR and N=S^L approaches for quantitative and symbolic sets. I've put forth two new number systems which may be significant. Virgil will discount all this, until someone he respects gives it some credit. Perhaps we can discuss those things in the future. I am only now getting a new web site begun, and will include papers on these things. In the meantime, thanks for your reflections.


If general set theory comes to a conclusion that contradicts the conclusions of a theory that takes into account more details of the situation, then hasn't the generalization failed in the specific case?

No. It just means there are many different notions of "size" that might apply to sets of some kind. A contradiction would occur only if the *same* mathematical notion of "size" led to different answers in the same situation. Now, you might wish argue that we should call cardinality something else than a "measure of the size of a set". But, as you surely realise, there is no hope of changing entrenched technical or informal terminology, nor is there really any reason to do so, simply because the only measure of size in abstract set theory that applies generally is cardinality.


Sure, it's an uphill battle to disagree with tradition. I didn't ask for a free ride. :)

The question is this. If cardinality DOES apply generally, then why does it contradict so many specific cases?

If I have a rule that all species of mammals produce milk and urine, then that should not have any exceptions, or it's not a general rule. If I have a rule that all mammals give live birth, well, that's generally true, but for a couple of exceptions. But if I have a rule that all mammals have four legs, we start to wonder how a good a rule it is, and if I say all mammals are nocturnal, we reject the rule. It's a matter of how many specific instances there are that contradict the rule

So again, the question is, in how many specific situations, where we have more information, do the general rules fail? I think, in any case where the infinite set is better structured and defined than raw set theory allows, the conclusions of abstract set theory fail, and are supplanted by better rules. This leads me to believe that trying to measure infinite sets without any additional structure is simply impossible. Perhaps this seems crankish, but I am hopeful that in some sense you can agree that there is an issue there.

:)

Tony


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