Re: Galileo's Paradox

Six wrote:
Only if '>' is left vague and undefined.

This is what gets me nettled straightaway. Surely this is something
to be established, rather than stated as a matter of fact at the outset.

If it's a question of whether a certain formal definition is
appropriate to capture an informal, intuitive notion, then yes, debate
can go on for a long time. But at a certain point, we'll want to draw
up a definition. If someone has another definition, then we can always
add that and then anyone is free to choose which he thinks better
captures the informal, intuitive notion.

Once we make clear definitions, we see that there is a difference
between one set being injectable (or injectable but not equinumerous)
into another and one set being a subset (or being a proper subset) of
another.

I understand the difference, but I'm not convinced of its relevance.

I don't know why it wouldn't be relevant. It seems to be at the crux of
what you're talking about.

Do you think it was ever a paradox, before these definitions were
dreamed up?

The word 'paradox' is itself informal. I do recognize that the fact
that the primes are 1-1 with the naturals struck (and still strikes)
people as paradoxical.

Galileo thought it a paradox, presumably. If it was ever a
paradox, do you not think there is the tiniest little scope for some very
clever mathematicians (not me) to debate how well the definitions measure
up to the original intuitions. Could be they were just all stupid back
then.

I think it's fine to see how things compare with intutions. I suspect
that just about anyone who has seriously studied set theory has thought
about the possibility of another definition along the lines you'd like
to see invented. Even I thought about that when I first studied some
set theory (and I'm only a beginning hobbyist-student of the subject).
But I don't know of such a viable proposal. It does seem to come down
to seeing that bijectability and proper subsethood are just two
different things.

Saying
that N = E says no more than that they are both infinite.

WRONG. Saying that N is equinumerous with E (not =) is saying there
exists a bijection between them. There are sets that are both infinite
but without a bijection between them.

You are jumping in here prematurely. The conceit here is that Cantor has
not yet arrived, if your imagination is up to it.

Okay, I'll play. But is this an historical question of what
mathematicians actually thought before Cantor, or a conceptual question
of how we would view this matter if we had not yet conceptualized the
basics of set theory of infinite sets?

If equinumerous is defined in the usual way, yes. It's a pity
though that word was appropriated. In a non-technical sense, meaning equal
in number, I would say it is not clear that N and E are equinumerous.

Other terms in use are 'equipollent', 'equipotent' and 'have the same
power'. Also, "bijectable" is used in these threads. I use
'equinumerous' usually, but any time I do use it, you may regard
'equipollent' to be used instead. 'equipollent' would be just as good
for me, except it's a bit old-fashioned.

They are all bijectable with one another. In Z set theory there is no
object that is the order of infinity for any cardinality in the sense
of every set of a certain cardinality being a member of that order.

I'm not trying to introduce any new concepts here. I'm just making the
distinction between the countable and uncountable infinities, and
suggesting that discoveries about the latter need have no direct bearing
on how the former are to be understood in themselves.

Okay, but I don't know what you have in mind specifically.

Imaginatorium has discussed below subsets and mappings, and there
are things I'm still trying to get to grips with here which may well make a
difference. But as I see things at the moments there is a primitive
intuition about size which is the same whichever we talk about.

Consider these two mappings from N to O (odd numbers):

the (non-injective) mapping:
1,2 -> 1
3,4 -> 3
5,6 -> 5
etc
and:

the (bijective) mapping:
1 -> 1
2 -> 3
3 -> 5
etc.


According to the first, we are inclined to think there are twice as
many members in N than O. According to the second mapping, we are inclined
to think there are exactly as many members in O as in N.
What gives the bijective mapping more weight?

Because it is indeed 'bi'; it goes both ways. (Ha! I didn't even intend
that to be a pun on sexuality.)

I would draw the conclusion that it makes no sense to think of N or
any denumerable set as having a fixed size. N is a piece of elastic, or
more accurately a piece of elastic with no resting state. We know it's
being stretched out to the rationals, and shrunk again when it goes down to
the even numbers, or squares. But there is no way of measuring these
stretchings. N is the ruler.

Okay. You might find a way to formalize that in set theory or some
other theory of your invention.

There are some radical consequences, which I think just have to be
accepted. We cannot say that N is base-invariant. On the other hand we
cannot say either that it has a different size for each base. Following a
nice argument from Hughes in zuhair's infinity thread (though I expect
Hughes to show me I've mangled it), we cannot even say that N is the same
size as itself,

I'm not familiar with the argument, but, yes, I don't imagine he argued
that we can't show that N does not have the same size as itself.

but fortunately we cannot say either that it has a
different size from itself (else we would be in trouble).

Indeed.

MoeBlee

.

Relevant Pages

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