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Topic: Infinity: The Story So Far
Replies: 298   Last Post: Apr 7, 2014 7:30 AM

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mueckenh@rz.fh-augsburg.de

Posts: 15,089
Registered: 1/29/05
Re: Infinity: The Story So Far
Posted: Feb 27, 2014 9:27 AM
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On Thursday, 27 February 2014 15:14:41 UTC+1, Dan Christensen wrote:
> On Thursday, February 27, 2014 6:54:55 AM UTC-5, muec...@rz.fh-augsburg.de wrote:
>

> > On Thursday, 27 February 2014 10:15:41 UTC+1, Virgil wrote:
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> > > Any set satisfying the Peano postulates can serve as "the" set of
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> > > natural numbers.
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> > Not in mathematics.
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> > This sequence satifies the Peano axioms:
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> > 0, 0.9, 1.8, 2.7, ...
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> > What axiom is violated?
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> The axioms only specify the name of the of the first number.


And the further numbers are not specified because logicians have never asked themselves whether other numbers than those known from childhood could follow. It is a pity that they cannot but imagine well-trodden paths.

> (Even for that, we have some leeway -- usually 0 or 1). In theory, you can choose whatever symbols you like for the rest.

Only if you know that they shall have the meaning known from childhood - not from Peano.


> You could, however, get into trouble in your example if you attach multiple meanings to the same symbol.

That would not be mathematics. But the other way round: you can attach different symbols like 1 and 1^1 and 1^1^1 to the same meaning. But that is not the most important failure. You do not get the natural numbers unless you specify that the differences must be units.

That is the indispensable property of the natural numbers - at least in real mathematics.

Regards, WM


Date Subject Author
2/25/14
Read Infinity: The Story So Far
Dan Christensen
2/25/14
Read Re: Infinity: The Story So Far
Brian Q. Hutchings
2/25/14
Read Re: Infinity: The Story So Far
ross.finlayson@gmail.com
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Read Re: Infinity: The Story So Far
ross.finlayson@gmail.com
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2/26/14
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3/1/14
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Dan Christensen
3/2/14
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Brian Q. Hutchings
2/26/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
2/26/14
Read Re: Infinity: The Story So Far
Dan Christensen
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Dan Christensen
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Martin Shobe
2/26/14
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ross.finlayson@gmail.com
2/27/14
Read Re: Infinity: The Story So Far
Dan Christensen
2/27/14
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Martin Shobe
2/27/14
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mueckenh@rz.fh-augsburg.de
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2/26/14
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mueckenh@rz.fh-augsburg.de
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2/26/14
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2/27/14
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mueckenh@rz.fh-augsburg.de
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mueckenh@rz.fh-augsburg.de
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3/5/14
Read Re: Peano's rot in action for you!
thenewcalculus@gmail.com
3/5/14
Read Re: Peano's rot in action for you!
Martin Shobe
3/5/14
Read Re: Peano's rot in action for you!
Wizard-Of-Oz
3/5/14
Read Re: Peano's rot in action for you!
Wizard-Of-Oz
3/5/14
Read Re: Peano's rot in action for you!
Wizard-Of-Oz
3/5/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
3/6/14
Read Re: Infinity: The Story So Far
Dan Christensen
3/6/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
3/6/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
3/11/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
3/11/14
Read Re: Infinity: The Story So Far
Wizard-Of-Oz
3/11/14
Read Re: Infinity: The Story So Far
Martin Shobe
3/11/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
3/11/14
Read Re: Infinity: The Story So Far
Martin Shobe
3/11/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
3/11/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
3/11/14
Read Re: Infinity: The Story So Far
Martin Shobe
3/11/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
3/11/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
3/11/14
Read Re: Infinity: The Story So Far
Martin Shobe
3/12/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
3/12/14
Read Re: Infinity: The Story So Far
Martin Shobe
3/12/14
Read Re: Infinity: The Story So Far
ross.finlayson@gmail.com

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