Re: Review of Mueckenheims book.

MoeBlee wrote:
On Mar 6, 5:25 pm, Tony Orlow <t...@xxxxxxxxxxxxx> wrote:
1. What does "arranged linearly" mean?

One after the other, sequential, like the naturals we were discussing
before someone threw kumquats into the equation.

See, it's good that we asked what your PERSONAL definition is,since
the usual definition of 'a linear ordering' does not require
sequencing, as 'sequence' itself has a mathematical definition.


Sure, that's fine. You don't consider a sequence to be possibly uncountable, and you consider a line to have an uncountable number of points, so those concepts are incompatible for you. However, if you consider Ross' (where is he, I wonder - on hiatus, too?) Finlayson numbers, an uncountable sequence of infinitesimally spaced iotas, as he calls them, the two concepts can be reconciled.

2. Why do you think "they" are arranged linearly?
Ask Peano.

Why not just look at any old proof that the standard ordering on
natural numbers is arranged linearly (even with your defintion of
'arranged linearaly')?


Why bother when it's obviously true? Shall I look up the definition of "of" too?

What's "infinite n"?
n| A meN n>m

What is that supposed to be?

n such that for all m in N, n>m


n is infinite <-> AmeN n>m.

Is what you're saying?


Yes, when you ask what "infinite n" is, I assume that you can accept a statement referring to the definition which is not a whole statement with the verb "implies". If you ask, "Where is the Babbo?" would you be confused if I said, "Under the sink", rather than, "The Babbo is under the sink"? I hope not.

What is YOUR definition of '>'?


That's a very good question, and one which has been among the fundamentals stewing in the back of my mind. Essentially, '<' is defined by:

a<b -> ~b<a

This does not imply that:

~b<a -> a<b

For it can be true that:

~b<a ^ ~a<b

or:

~(b<a v a<b)

in which case we deine '=':

~(a<b v b<a) <-> a=b

Since we can exchange a and b in this statement without changing it we have a reflexive function between them:

a=b <-> b=a

We can also define '>' as:

a<b <-> b>a

And that's about all there is to say about '<', from a language standpoint.

Of course, this really only applies where we have some space with a dimension along which a<b, a=b, or a>b. This is a linear space, e.g. the number line, and '<' can be interpreted geometrically as "below" or "to the left" or "in the negative direction from".

You can't explain what "infinite-case induction" is because you can't
explain what "infinite numbers" are.

I did, in the same basic way Robinson did, as a number greater than any
finite number.

'infinite number' in that sense is distinct from 'is infinite' in the
sense of cardinality.

Yes, that's true. Set theoretic infinity is defined using bijection with a proper subset, which includes countable infinities, which this definition really does not.


But good to see you appreciate A. Robinson, whose work is built on
classical mathematics, including mathematical logic, ZFC, and is non-
constructive up the wazoo.

On the other hand, it's hard to make sense of your endorsement of
Robinson's non-standard analysis - built on classical and decidedly
non-constructive mathematics and ZFC, in the sense of using the
principles of ZFC including Zorn's lemma - when you claim ZFC is all
wrong.

MoeBlee


I claim that pure sets cannot be infinite, and that set theoretic infinity depends on something other than set membership, even if an inductive set can be defined using only 'e'. Where it is, each element is a proper superset of the last, and so in that sense, there is a sequence, which is more than a set, and it depends in one way or another of successive elements growing in "size" in one way or another. It's not that set theory is "wrong" persay, just that it extends its claims of truth beyond its actually capabilities, where it ignores measure. Of course, you would probably say that about me... :)

ToeKnee
.

Relevant Pages

  • Re: Well Ordering the Reals
    ... I have always allowed that sequences can be countably infinite, ... like the sequence of standard naturals is. ... TO nor anyone else has produced any axiom system in which it can be ... at several removes from pure mathematics, ...
    (sci.math)
  • Re: Review of Mueckenheims book.
    ... as 'sequence' itself has a mathematical definition. ... explain what "infinite numbers" are. ... classical mathematics, including mathematical logic, ZFC, and is non- ...
    (sci.math)
  • Re: No Unique Initial Segment And No Characteristic Expansion.
    ... >> Infinite people each flip coins infinite times. ... >> Can you always find a different sequence of heads and tails? ... During the rush to reformalize mathematics in the 20th century, ...
    (sci.math)
  • Re: No Unique Initial Segment And No Characteristic Expansion.
    ... >> Infinite people each flip coins infinite times. ... >> Can you always find a different sequence of heads and tails? ... During the rush to reformalize mathematics in the 20th century, ...
    (sci.logic)
  • Re: No Unique Initial Segment And No Characteristic Expansion.
    ... >> Infinite people each flip coins infinite times. ... >> Can you always find a different sequence of heads and tails? ... During the rush to reformalize mathematics in the 20th century, ...
    (sci.physics)