A relative failure

Originally posted by ChronicLeaky
A serious book on set theory will say something like "The asymptotic density of the even integers in the set of integers is [b]one half". I still don't like Dr S's use of the verb "halve" before, though, because when we halve a number, we get a unique result, while "halving" the integers by constructing some set with density 1/2 can be done in unco ...[text shortened]... concepts on the basis that the same word sometimes gets used for both.[/b]

That hurt! That hurt really really really bad! Oh, man how that hurt. 🙁
Kelly

Originally posted by ChronicLeaky
A serious book on set theory will say something like "The asymptotic density of the even integers in the set of integers is [b]one half". I still don't like Dr S's use of the verb "halve" before, though, because when we halve a number, we get a unique result, while "halving" the integers by constructing some set with density 1/2 can be done in unco wo things "Al Gore linguists" in honour of Al Gore's claims of environmentalism.[/b]

Come on, every mathematician would be an Al Gore linguist, as all of mathematics is rife with overloaded terms.

Matrix multiplication is different from real multiplication, but you don't take issue with overloading that term, do you? Hell, even imaginary multiplication is different from real multiplication.

But the very reason such terms are overloaded is at the essense of mathematics: the power of abstraction. It is the very insight gained from examing commonality among things that look superfically different, like halving numbers contrasted with halving sets, that makes math more interesting than grade school exercises.

Further I made explicit what I meant by halving a set: constructing two disjoint sets of equal cardinality whose union is the original set. My halving refers to one thing, even though it has multiple solutions, just like the square root operation refers to one thing while having multiple solutions (or factoring, or finding maxima in a function's domain, etc.).

Originally posted by DoctorScribbles
Let us make the following wager.

An infinitely long random integer will be generated, one random digit at a time. For each digit that is even, you pay me $3. For each that is odd, I pay you $1.

What do you say?

I did say that you could put an infinite number on top of the infinite
didn't I? Even numbers or odd are still an infinite amount of numbers
are they not?
Kelly

Originally posted by KellyJay
I did say that you could put an infinite number on top of the infinite
didn't I? Even numbers or odd are still an infinite amount of numbers
are they not?
Kelly

They are. Each set has infinite cardinality -- there are no more integers than there are even integers. However, exactly half of the integers are even.

Originally posted by DoctorScribbles
Come on, every mathematician would be an Al Gore linguist, as all of mathematics is rife with overloaded terms.

Matrix multiplication is different from real multiplication, but you don't take issue with overloading that term, do you? Hell, even imaginary multiplication is different from real multiplication.

But the very reason such terms are ...[text shortened]... hile having multiple solutions (or factoring, or finding maxima in a function's domain, etc.).

True, to a point, but there isn't enough of an abstract connection between the two uses of "halve" to justify this. I mean, matrix multiplication is a function from some general linear group crossed with itself to itself, and multiplication is a function from some Cartesian product of two sets to one of the sets. The similarities are very clear; in particular, they're both operations which are well-defined, by virtue of being functions.

Halving numbers and halving sets are not even superficially similar. Halving a number is a function from a division ring to itself. Halving a set is something which, as I showed above, needs to be defined separately in uncountably many cases. The overloading of this term doesn't have any benefits I can think of, abstractionwise. Instead, as KellyJay showed, it only serves to confuse issues.

See my edit, though. I don't have a beef with your halving anymore.

Originally posted by ChronicLeaky
The overloading of this term doesn't have any benefits I can think of, abstractionwise.

I would argue that set halving corresponds at least as closely to "real world" halving as does number halving. They regularly go hand in hand.

If we evenly share a $10 pizza, we halve the number 10 to find that we should each owe $5. But we physically partition the pizza slices into two sets, which can be done in a variety of custom-defined ways. Maybe I take the left half, or every other slice, or the half with mushrooms, or we enumerate 6 portions and roll a die, etc.

The abstraction is one of a particular sort of complementariness, in which both complements contribute in some sense equally to the whole. We should each get half of the set of slices because we each paid half; or, we should each pay half because we each got half of the set of slices. If you argue that there is no underlying abstraction connecting these two facets of pizza sharing, on what basis would you determine each person's share of the price or the slices? The benefit of the overloaded term is that when we agree to split a pizza, we both know we'll be paying for half of it and eating half of the slices, without having to stipulate each individually, and the details of the slice set halving are not germane to the deal.

Originally posted by DoctorScribbles
Further I made explicit what I meant by halving a set: constructing two disjoint sets of equal cardinality whose union is the original set.

Would constructing these two sets mean halving the set of integers by your definition or do you need some correspondence (and not simply infinite cardinality in both sets)?

Set1: {All integers greater than -100}
Set2: {All integers lesser or equal than -100}

Originally posted by Palynka
Would constructing these two sets mean halving the set of integers by your definition or do you need some correspondence (and not simply infinite cardinality in both sets)?

Set1: {All integers greater than -100}
Set2: {All integers lesser or equal than -100}

Sure, those two sets halve the integers.

There is in fact a one-to-one correspondence between their elements.

Originally posted by DoctorScribbles
Sure, those two sets halve the integers.

There is in fact a one-to-one correspondence between their elements.

Can you elaborate on that correspondence?

Originally posted by Palynka
Can you elaborate on that correspondence?

Let x be any member of the second set.

Define y to be (-x - 199).

Then y will be a member of the first set corresponding to x, and for every element y in the first set, there is an x in the second set to which it corresponds.

In simple terms, this is the absolute value, relative to -100, correspondence.

Originally posted by DoctorScribbles
They are. Each set has infinite cardinality -- there are no more integers than there are even integers. However, exactly half of the integers are even.

Like I said, there are a lot of things we can say and do; my point
is that using math and math alone to look at the universe, we will
still miss things. Not registering an ink dot on an infinitely large
piece of paper in size with percentages that is forever growing in
size should show us that time before the Big Bang is simply
something not seen through a mathematical model yet could still
exists.
Kelly

Originally posted by KellyJay
my point
is that using math and math alone to look at the universe, we will
still miss things.

Oh, so that was your point? Sorry, but I really couldn't figure it out from your previous posts. Well, if that's your point, I agree, but I don't really see the point with your point. Is there anyone who thinks that maths and maths alone can explain the universe?

Originally posted by DoctorScribbles
Let x be any member of the second set.

Define y to be (-x - 199).

Then y will be a member of the first set corresponding to x, and for every element y in the first set, there is an x in the second set to which it corresponds.

In simple terms, this is the absolute value, relative to -100, correspondence.

Nice. I was having a problem with zero in finding an even/odd correspondence, but now I see how one can use a similar reasoning.

Originally posted by Nordlys
Oh, so that was your point? Sorry, but I really couldn't figure it out from your previous posts. Well, if that's your point, I agree, but I don't really see the point with your point. Is there anyone who thinks that maths and maths alone can explain the universe?

Can you see how time before the Big Bang could be real and missed?
Kelly

Originally posted by ChronicLeaky
I don't have a beef with your halving anymore.

All of a sudden, I do.

I just realized that according to me, the set of integers which are multiples of 3 (or 4, 5, 6, 7...) constitutes half of the set of integers, complemented by the set of non-multiples. Applying my analysis of the proposed wager, I'd have to accept an even odds wager on drawing a multiple of 7 at random. But I'd obviously expect to lose at such a wager, so that analysis is crap under my definition of halving a set.

It seems that there ought to be a way to express that the even numbers constitute half of the integers in a way that the multiples of 3 don't, but my definition of set halving clearly fails at that. Does the notion of asymptotic density suffice to remedy this?

God damn infinite sets.