Where definitions end - Page 3

smrk2 · 07-07-2012, 11:45 PM

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Originally Posted by lastcardcharlie

Except that sets precede numbers, at least in the modern view. For example, the Peano formulation of the natural numbers consists of a set together with a successor function.

Just a flesh wound!

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But, if you limited the elements of a set to strings of symbols of the form, say: a, aa, aaa, ..., (some, none, or all of them, and without a successor function, which would turn them into natural numbers) that would in principle be fine by me, although most inconvenient in practice. Expressed more formally, this is the view that all sets of the same cardinality are isomorphic, i.e. there is no structural difference between them at all, and it does not matter what their elements are.

So for the sake of argument, limiting the elements of sets to strings of symbols, do you still think that 'element of a set' or 'set' is undefinable?

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Could be, not sure what Cantor meant, although the elements of a set do all have to be different, of course.

It seems that you have a problem primarily with the phrase 'distinct objects of our perception'. It may sound like arcane weirdo phenomenology, but I don't see any deep significance in the 'of our perception' part; I think he's just saying that we can treat apples and oranges (of our perception) as elements of sets or we can treat numbers and sets (of our thought) as elements of sets. Maybe I'm pretty naive on this turf though.

Aaron W. · 07-07-2012, 11:57 PM

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Originally Posted by BruceZ

The official position seems to be that if I ask you to prove something about {x,y}, and your proof assumes that x does not equal y, then your proof is invalid.

I disagree with this. "Officially" (is there an actual official for this?) I would say that |{x, y}| = 2. I don't think it makes sense to say |{x, y}| = 1 under any "official" circumstances. It's either a set containing two distinct objects, or a multiset containing the same object twice.

Unofficially, if someone really wanted to make a fuss, the official line could be easily modified to accommodate it. But I would say that this is shallow and pedantic.

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So if we can have {x,y} when x = y, then what about {x,y} when x ≡ y? You seem to be saying the same thing about = as ≡ which means identically equal, in which case x and y can be identical objects. For example, do you consider

{1.9999...., 2}

valid? Those are identically equal. They refer to the same object.

Yes and no. They refer to the same object in the context of the real numbers. For example, they define two distinct infinite sequences of digits. It's only under an equivalence relation that we say that they "refer to the same object."

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If it's just a question of notation, it isn't interesting, but if you are saying that I can't conceptualize the set consisting of a and a, then I disagree. I can conceptualize that just fine. Now I'm going to go eat dinner and go eat dinner.

I would say that you're conceptualizing a multiset.

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Originally Posted by Cantor

A set is a gathering together into a whole of definite, distinct objects of our perception

a and a are not distinct objects of our perception. They're the same object of our perception.

Aaron W. · 07-08-2012, 12:06 AM

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Originally Posted by smrk2

Maybe I'm pretty naive on this turf though.

Given that most of mathematics can be pursued using naive set theory (that is, a lot of mathematicians NEVER explicitly invoke the axioms of ZF/ZFC and don't operate with objects that create the pathologies), I think you're in good company.

**BruceZ** · 07-08-2012, 12:56 AM

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Originally Posted by Aaron W.

I disagree with this. "Officially" (is there an actual official for this?) I would say that |{x, y}| = 2. I don't think it makes sense to say |{x, y}| = 1 under any "official" circumstances. It's either a set containing two distinct objects, or a multiset containing the same object twice.

Unofficially, if someone really wanted to make a fuss, the official line could be easily modified to accommodate it. But I would say that this is shallow and pedantic.

Well that's exactly what I argued, and I was steamrolled by the emeritus algebraist, and Jason1990 stated the same thing. That official enough for me. About the only one who agreed with me was you.

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Yes and no. They refer to the same object in the context of the real numbers. For example, they define two distinct infinite sequences of digits. It's only under an equivalence relation that we say that they "refer to the same object."

OK. How about {pi/4, arctan(1)}?

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I would say that you're conceptualizing a multiset.

It's not a multiset because there is no distinction made for position.

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A set is a gathering together into a whole of definite, distinct objects of our perception

a and a are not distinct objects of our perception. They're the same object of our perception.

Well {a} is certainly not a gathering together of distinct objects.

Aaron W. · 07-08-2012, 01:24 AM

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Originally Posted by BruceZ

Well that's exactly what I argued, and I was steamrolled by the emeritus algebraist, and Jason1990 stated the same thing. That official enough for me. About the only one who agreed with me was you.

So I'm not official enough?

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OK. How about {pi/4, arctan(1)}?

If we are thinking of these as values, I would say it's not valid. If we are thinking of these as just collections of symbols (that is, not endowing them with numerical meaning, but as literal strings of characters), then it's perfectly valid. Again, it matters how we're viewing the objects. (And we can never add a condition like that with a and a. Those are ALWAYS the same.)

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It's not a multiset because there is no distinction made for position.

A multiset is just a set that is allowed to contain multiple copies of the same object. It sounds like you're thinking of a tuple (which is an ordered list).

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Well {a} is certainly not a gathering together of distinct objects.

If you're going to go that route, I'm surprised that you didn't go with the empty set. Or are you really talking about the pluralized form of object? Either way, it's as much of a gathering of objects as the following is a sum of terms:

$\sum_{n=1}^{1} n$

lastcardcharlie · 07-08-2012, 05:35 AM

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Originally Posted by smrk2

So for the sake of argument, limiting the elements of sets to strings of symbols, do you still think that 'element of a set' or 'set' is undefinable?

In the OP I was stating what I believed to be the standard view that sets are undefined. This has always baffled me, hence this thread. If you restrict elements to strings of the specific symbols a_1, ..., a_n, the conclusion that sets are now defined seems hard to avoid. However, that would also seem to restrict sets to having cardinality at most that of the reals (doesn't matter to me, but unacceptable to set theorists, of course).