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/sci/ - Science & Math

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9523101 No.9523101 [Reply] [Original]

Why are numbers defined
0 = ()
1 = (0) = (())
2 = (0,1) = ((), (()))

And not
0 = ()
1 = (0) = (())
2 = (1) = ((()))

>> No.9523217

>>9523101
It means that the peano successor function is just the powerset and the 'value' of the number is just the cardinality. Also means that the subset relation is an ordering on the integers just like <

>> No.9523247
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9523247

>using parentheses for sets

>> No.9523258

>>9523101
They can be defined that way too.
The advantage of the Von Neumann definition is that it generalizes to ordinals such that each ordinal is the set of all previous ordinals.

You can't "take the limit" of the other one because then what would be the elements of [math] \omega [math]?

>> No.9523265

>>9523217
Successor isn't powerset, it's [math]x \mapsto x \cup \{x\}[/math]

>> No.9523272

>>9523101
The cardinality of each number is the number itself this way.

>> No.9524098

>>9523272
Thanks, good answer! I understand it now