Thank your for the astute response.
1.You say that the points are brutely distinguishable and later you say that they are indistinguishable, which nevertheless you hold to be different properties.
The points are brutely distinguishable, but the sets aren’t.
2.Why are the sets indistinguishable? Although I don’t particularly understand what predicates you allow for brutely distinguishable entities, it seems possible to have X = set of all brutely distinguishable points (from some class) and Y = set of all brutely distinguishable points except one. It is, of course, not a definition of Y unless you point out which of the points is missing (which you presumably can’t), but even if you don’t have a definition of Y, Y may still exist and be distinguishable from X by the property that X contains all the points while Y doesn’t.
No predicates besides brute distinguishability govern it. Entities that are brutely distinguishable are different only by virtue of being different.
The sets that differ but for one element differ because their cardinality is different. This is how they differ from the infinite case.
3.If the argument were true, haven’t you just shown only that you can’t define an infinite set of brutely distinguishable entities, rather than that infinite sets can’t be defined at all?
If infinite sets and brutely distinguishable elements exist, infinite sets with brutely distinguishable elements should exist.
4.What is your opinion about the set of all natural numbers? Is it finite or can’t it be defined?
It is infinite, but it isn’t “actually realized.” (They don’t exist; we employ them as useful fictions.)
And how does the argument depend on infiniteness, after all? Assume there is a class of eleven brutely distinguishable points. Now, can you define a set containing seven of them? If you can’t, since there is no property to distinguish those seven from the remaining four, doesn’t it equally well prove that sets of cardinality seven don’t exist?
To make the cases parallel (which I hope doesn’t miss the point): take 7 brutely distinguishable points; 4 more pop into existence. The former and latter sets are distinguishable by their cardinality. When the sets are infinite, the cardinality is identical.
The frequentist definition of probability says that probability is the limit of relative frequencies, which is the limit of (the number of occurrences divided by the number of trials), which is not equal to (limit of the number of occurrences) divided by (limit of the number of trials).
This doesn’t seem relevant to actually realized infinities, since the limit becomes inclusive rather than exclusive (of infinity). The relative frequency of heads to tails with an actually existing infinity of tosses is undefined. (Or would you contend it is .5?)
And emphasis is usually marked by italics, not red.
Are my aesthetics off? I’ve decided that unbolded red is best for emphasizing text sentences, the reason being that it is more legible than bolding, centainly than italics. If you don’t use many pictures or diagrams, I think helps maintain interest to include some color whenever you can justify it.
Color seems increasingly used in textbooks. Perhaps its a status thing, as the research journals don’t use it. But blog writing should usually be more succinct than research-journal writing, and this makes typographical emphasis more valuable because there’s less opportunity to imply emphasis textually.
I just found it curious: I’ve addressed typography issues in a blog posting, “Emphasis by Typography.”
I have to say I’m surprised by your tone; like you’re accusing me of some form of immorality for not being attentive to readers. This all strikes me as very curious. I read Hanson’s blog and so have gotten attuned to status issues. I’m not plotting a revolution over font choice; I’m only curious about why people find Verdana objectionable just because other postings use a different font.
The argument concerns conceptual possibility, not empirical existence. If actually existing sets can consist of brutely distinguishable elements and of infinite elements, there’s nothing to stop it conceptually from being both.
You have located a place for a counter-argument: supplying the conceptual basis. But it seems unlikely that a conceptual argument would successfully undermine brutely distinguishable infinite elements without undermining brutely distinguishable elements in general.
You can distinguish the cardinality of finite sets with brutely distinguishable points. That is, if a set contains 7 points, you can know there are seven different points, and that’s all you can know about them.