Re: infinity

Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> stephen@xxxxxxxxxx said:
>> Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
>> > Daryl McCullough said:
>> >> Tony Orlow writes:
>> >>
>> >> >Do you think I was trying to prove that any infinite set must
>> >> >have elements with an infinite number of
>> >> >predecessors? That should be obvious. Is it not?
>> >>
>> >> No, it's not obvious. The *negation* of that statement is obvious
>> >> to most people.
>> >>
>> >> --
>> >> Daryl McCullough
>> >> Ithaca, NY
>> >>
>> >>
>> > Only because of what they have been told, repeatedly, which is wrong.
>>
>> Where do you get your definitions from Tony? You do realize
>> that words have no inherent meaning. Word meaning is a
>> matter of social convention. You seem to think there is
>> some "real" definition of infinite that somehow differs from
>> the mathematical definition, and also from the naive man-on-the-street
>> never ending definition.
>>
>> So on what basis do you claim that the mathematical definition
>> of 'infinite' is wrong?
>>
>> On what basis do you claim that the "without end" definition
>> of 'infinite' is wrong?
>>
>> Stephen
>>
> On the basis of the contradictions that arise when we apply known properties of
> finite and infinite numbers. While "infinite" means "without end", and thus the
> set of naturals as defined by Peano is therefore infinite, the additional
> restriction that the elements be finite, and therefore have some unknown "end"
> to their values, implies that the set is not infinite as defined, but has some
> unknown "end" to its elements.

There are no contradictions with the standard definitions.
Claiming that the standard definitions contradict your
ideas is irrelevant.

Each natural number is finite. Each natural number is bounded.
For each natural number n, the set
{1, 2, 3, 4, ... n}
has a last element.

The set of all finite natural numbers is not finite. There
is no finite bound that is larger than each and every finite
number. The set
{1, 2, 3, 4, ....}
does not have a last element.

You apparently agree with these statements, yet
you insist there is some contradiction.

When talking about lists of positive integers, the mathematical
definition of finite is essentially equivalent to 'has
a last element', and the definition of infinite is
'does not have a last element'.

Consider the list of lists
L = { {1},
{1, 2 },
{1, 2, 3 },
.....
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 },
.....
{1, 2, ...., 9999, 1000 },
.....
{1, 2, ...., 235989325, 235989326, 23598327 },
.....
}

Each element of L is a list that has a last element.
Every element of L is a list of the form {1, 2, 3, .... n}
where n is a finite number. The list L itself does
not have a last element.

Do you claim this is a contradiction? If you
do not think this is a contradiction, then why
do you have a problem with an infinite set of finite
naturals?

Do any of the elements in L not have a last element?
Given that they are all defined to have a last element,
they all must have a last element, even the non-existent
ones, according to your logic.

Does L have a last element? I suppose you will
claim it tenuously exists, whatever the hell that
is supposed to mean.

Stephen
.

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