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Topic: Warning to visitors to sci.math
Replies: 266   Last Post: Mar 19, 2012 9:13 AM

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Rotwang

Posts: 1,685
From: Swansea
Registered: 7/26/06
Re: Warning to visitors to sci.math
Posted: Mar 16, 2012 12:24 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On 16/03/2012 03:43, DavidW wrote:
> Rotwang wrote:
>> On 16/03/2012 02:52, DavidW wrote:
>>> [...]
>>>
>>> Did you need to use infinite sets or only induction?

>>
>> Infinite sets, specifically the sets s_n, as well as {lub s_n | n in
>> N} and {glb s_n | n in N}. Induction doesn't come into that part at
>> all and I don't see how it could

>
> I'll go away and have a closer look at your proof and stop bugging sci.math with
> this for the moment. But before looking at it my thinking is that infinite sets,
> including your s_n etc. are defined by induction.


I don't know what that could mean.


> The question is whether you
> need "all" elements of those sets at any stage to make the proof work, or if you
> only need to prove that particular properties hold for each successive element
> in them. I would be troubled if you really need "all" of an infinite set,
> because I consider that a contradiction in terms.


Well, you could change the real number axioms by replacing the
completeness axiom with an axiom that states that every Cauchy sequence
has a limit (which can be proved to be equivalent to completeness if one
uses infinite sets), for example - that would bypass the need for
infinite sets in the proof I gave. But I don't see what you would gain
from this, since I don't see why you think that the notion of an
infinite set is any more problematic than that of an infinite sequence.
You've said that one "cannot have _all_ of something that is unbounded".
Why do you think that the existence of the set of natural numbers means
having all of something unbounded, but the existence of an infinite
sequence (that is, a function whose domain is all of the natural
numbers) doesn't? Is it just that the word "set" invokes the image of a
container, and you don't accept the idea of something containing an
infinite number of things? If so then just call them something other
than sets, and use a word other than "in" for the elementhood relation.
In practice, nothing that mathematicians use sets for actually depend on
thinking of sets as "containing" things.


Date Subject Author
3/7/12
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DavidW
3/7/12
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adamk
3/8/12
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Brian Chandler
3/8/12
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Richard Tobin
3/8/12
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Shmuel (Seymour J.) Metz
3/8/12
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amzoti
3/9/12
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micky
3/8/12
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Frederick Williams
3/8/12
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DavidW
3/8/12
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DavidW
3/8/12
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DavidW
3/8/12
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J. Antonio Perez M.
3/8/12
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DavidW
3/8/12
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J. Antonio Perez M.
3/9/12
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DavidW
3/9/12
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Brian Chandler
3/9/12
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DavidW
3/9/12
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DavidW
3/9/12
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Brian Chandler
3/9/12
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DavidW
3/9/12
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Frederick Williams
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DavidW
3/9/12
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Frederick Williams
3/9/12
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DavidW
3/9/12
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J. Antonio Perez M.
3/10/12
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Frederick Williams
3/10/12
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DavidW
3/10/12
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DavidW
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3/10/12
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3/12/12
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Tom P
3/12/12
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3/12/12
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3/12/12
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3/12/12
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DavidW
3/13/12
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Rotwang
3/13/12
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DavidW
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Rotwang
3/13/12
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DavidW
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Rotwang
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DavidW
3/13/12
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Rotwang
3/14/12
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DavidW
3/14/12
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Rotwang
3/14/12
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J. Antonio Perez M.
3/14/12
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DavidW
3/14/12
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Horand.Gassmann@googlemail.com
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J. Antonio Perez M.
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DavidW
3/14/12
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DavidW
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Rotwang
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Rotwang
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Transfer Principle
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Daryl McCullough
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3/14/12
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Stan Brown
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R H Draney
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3/13/12
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3/13/12
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J. Antonio Perez M.
3/13/12
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Richard Tobin
3/13/12
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Peter Moylan
3/14/12
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R H Draney
3/14/12
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Transfer Principle
3/14/12
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Rotwang
3/15/12
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Rotwang
3/14/12
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ross.finlayson@gmail.com
3/14/12
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J. Antonio Perez M.
3/15/12
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Transfer Principle
3/16/12
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Daryl McCullough
3/17/12
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Michael Stemper
3/17/12
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3/14/12
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Michael Stemper
3/14/12
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Tim Little
3/13/12
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3/13/12
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3/14/12
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3/14/12
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3/15/12
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3/16/12
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quasi
3/16/12
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3/15/12
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3/10/12
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3/12/12
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3/13/12
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3/13/12
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3/10/12
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3/11/12
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3/12/12
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3/8/12
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3/10/12
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3/10/12
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3/10/12
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3/11/12
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3/11/12
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3/13/12
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