Re: Infinite numbers. An alternative to Cantor.

Mitch Harris wrote:
On Dec 17, 3:02 pm, John Jones <jonescard...@xxxxxxx> wrote:
Mitch Harris wrote:

> On Dec 16, 7:16 pm, John Jones <jonescard...@xxxxxxx> wrote:
>> 1) 2 is infinite because 2 has been given no successor.

> I'm not sure what you mean by infinite here. 2 has an obvious
> successor,

Not in this case.

What case? You're thinking of a particular concept of 2 that you have
or to the general one that most people have?

The case where a termination of a calculation, a finish, is 2. See next sentence.

There are no successors of a number that is a product
or termination of a calculation or presentation. If 2 had a successor
then I could not write simply 2.

Pardon my naivete, but 3 doesn't work for you?

No, 3 doesn't work. If 3 followed 2 necessarily then a product of one and one would be three.

The only situation where 2 has a successor is in a sequence.

The sequence is defined by the successor relation, not the other way
around.

Yes, we are talking about the same idea.

I can't
represent a solitary number from a sequence without also representing
its successor.

same thing, cart before the horse.

Yes, cart, horse, whatever.

That is, a sequence (of successions) is always given in
its entirety.

and that doesn't follow ("That is" usually precedes an explanation,
and that wasn't one)
The rule of succession might be given, but that's not the same as the
entire sequence itself.

I think it is. Otherwise a sequence is not generated by succession.


Therefore, if I want a product or termination then I need to abandon
succession.

- "Product or succession"? You use those as though they are synonyms,
but I don't see it.

You do see what I mean though.

- "Therefore": how does that follow? I don't see it.

If I want to terminate a mathematical procedure (eg in working out a quadratic equation) then I must abandon succession at the point of termination.

- why do you -need- to abandon succession? What does that even mean?

I must abandon the rule of succession if I want to end my calculation. If I do not abandon the rule of succession then the termination of a calculation is the sequence of natural numbers.

Now the number that is not succeeded in a succession is
infinite.

No, that's wrong use of the word 'infinite'. A number that is not the
minimum and has no -predecessor- (cannot be reached by succession) is
infinite. Hm...I don't think that is a definition, just a property
that can be inferred (aleph_null, the number of natural numbers has
that property, but so can other numbers

That is why 2, on its own, is infinite,

What? Sure, if you define 'air' to be 'chair', you sit on air.

But 2 -does- have a successor, as can every natural number (by
definition?).

Yes, 2 does have a successor if we say it has. We say 2 has a successor when we want to create a sequence of which 2 is necessarily a member. But 2 that is not in a sequence has no successor.

aside from those cases where it is merely a numeral.

2 isn't a numeral...oh never mind.

Yes, never mind that one.

> so I'm not sure what you can mean by 'has been given'. Even
> if you stipulate no successor to 2, I'm guessing you will allow that 2
> is the successor to 1, which is the successor to 0 (which is given),
> and so 2 is finite.

Yes. But that is true only for numbers in sequences. Numbers that arise
as products or terminations of calculation are not in a sequence.

What? How do you know that? So let's take the calculation 17 - 5. It
then produces 12. 12 is still the end of a sequence starting at 0 and
using successor a few times (and of course one can continue to
others).

Yes. 12 can be placed at the end of a sequence 0 to 12. 12 is also infinite in this case as it has no successor.



>> 2) 2 is countable because counting requires a termination, and 2 is a
>> termination.
>>
>> Accepting this initial conclusion
>
> I'm not sure what you have concluded.

A calculation requires us to stop at some point. That stopping point is
the termination of the calculation.

An end to a succession of manipulations? So the number is the end of a
sequence of manipulations?

You know what I mean. If teacher asks us to work out a quadratic then we need to know when to stop calculating. Remember, mathematics does not tell us the start and end of calculating. If we want to mathematically indicate the termination of a calculation then numbers must have properties that tell us when a calculation is terminated. Yes?

A number associated with a
termination is the largest conceivable number. 2 and 2 is 4. 4 is the
terminating digit. A number that is a terminator has no succeeding
numbers. If it did, these would have to be represented as well.

and 5 doesn't have a representation?

If I work out a sum and the answer is 4 then I would have no business claiming that 4 in my calculation has a successor.

> That 2 is both finite and
> infinite? No, you're then talking about two (!) different kinds of
> '2'.

Yes. Numbers in a sequence have different properties to numbers that are
not in a sequence. Numbers in sequences cannot be used in calculation.
Numbers are not transferable between applications in any case.

What? Really? That's crazy. One can look at the sequence property or
ignore it, the different properties can be used in different
circumstances, but still have both.

Numbers are not transferable between applications (there is no universal frameowrk for representing numbers). What is transferred are empty signs, or numerals.

>> would bring us to the idea that in
>> mathematics today there are two non-quantifiable ways of expressing
>> infinities:
>> a) presentable
>> b) not presentable.
>> Digits, like 2, 36, 765, etc., fall in group a)
>> Symbols, like aleph-null, fall in group b)
>
> You've made a big leap here. You've taken it as obvious that your
> discussion of '2' leads to a useful concept called 'presentability',
> you gave some examples, but it is not at all clear what it means. Why
> isn't aleph_null presentable?

Aleph null isn't presentable as a digit.

So what? Why is that problem? A number greater than 9 can be
represented as seqences of digits 0 through 9, -or- by a symbol we
choose ('A' as in hexadecimal notation).

I can't do with a symbol what I can with a digit. I can't claim that two unknowns in a quadratic are known, for example. I have to show that they are known and that is achieved by representing them as digits.

We may assume otherwise, but
that expresses a difficulty not of my doing. The difficulty of
representing aleph null as a digit is a necessary difficulty, and not
simply a possible difficulty.

So for the Piraha in the Amazon (a small group whose language has mo
term for 3), representing 3 is a -necessary- difficulty?

You are claiming that there is an identity between two numbers if they are conceptually linked as digit and symbol. I might be able to symbolise three, but would I be able to work quantitatively with it?

> and presentability doesn't seem to have
> anything at all to do with infinity.

Infinity lays claim to number, but as a digit it is necessarily
unrepresentable.

No not necessarily. Here is a representation: aleph_null.

It isn't a quantitative representation. It's a concept. It doesn't do the things that a quantitative representation can do.


> And your distinction seems to be applied just to infinities, but then
> you contrast in our examples finite #'s (in the standard definition of
> finite) with an infinite one. It sounds like your definition of
> 'presentable' boils down to the standard definition of 'finite' (not
> your 'finite' but other people's 'finite')

Presentable would be a number expressible as digits.

Oh. That's it? There's not much special about digits.

Yes, the presentation of quantity, rather than its conceptual representation.

Mitch
.

Relevant Pages

  • Re: Infinite numbers. An alternative to Cantor.
    ...  > I'm not sure what you mean by infinite here. ... If 2 had a successor ... The sequence is defined by the successor relation, ... Aleph null isn't presentable as a digit. ...
    (sci.logic)
  • Re: infinity
    ... Not while discussing the consequences of the axioms as they actually are! ... > You never commented on my adjustment of the Peano axioms, ... IF you claim to have generated an infinite set with your ... I am applying only finitely many successor operations at ...
    (sci.math)
  • Re: infinity
    ... >>> The Peano axioms prohibit any complete Peano set from having a largest. ... You never commented on my adjustment of the Peano axioms, ... Well, after all, you ARE applying an INFINITE number of successor operations, ... IF you claim to have generated an infinite set with your stepwise difninition, ...
    (sci.math)
  • Re: Infinite numbers. An alternative to Cantor.
    ... > I'm not sure what you mean by infinite here. ... or termination of a calculation or presentation. ... If 2 had a successor ... represent a solitary number from a sequence without also representing ...
    (sci.logic)
  • Re: Infinite numbers. An alternative to Cantor.
    ... The case where a termination of a calculation, a finish, is 2. ... If 2 had a successor ... calculation is the sequence of natural numbers. ...
    (sci.logic)