Varieties of Successorship (Set Theory)

A video explaining that there are different ways to define the Peano concept of successorship in Set theory, following either Von Neumann or Zermelo, as well as the notation of successorship.
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[@RENATVS_IV]

The comments below on the definition of 0 are very interesting.
I don't fully understand why the definition must be the "empty set" when the cardinality of a set is the important property in which you use the numbers, such as 0

[@ostrich_dog]

think about what it would mean for a set to contain a number. a set cannot contain a cardinality, but it can contain other sets, so it makes sense to define numbers as sets

[@InventiveHarvest]

I can define the word "elephant" as a reptile with no legs, but that doesn't mean I should. Typically, "zero" is defined as a number with no value. Its going to be hard to derive the second postulate (numbers follow numbers) if zero is a set and not a number.

[@Overonator]

0 is a number and it's defined as the empty set.

[@InventiveHarvest]

@@Overonator a number is a value or amount. A set is a collection of distinct entities.

[@Overonator]

@@InventiveHarvestOkay then zero has the no value or no amount. It's a set with nothing in it.

[@InventiveHarvest]

@@Overonator incorrect. There is exactly one empty set. Zero applies to the amount of entities IN the empty set, but not the set itself.

[@Overonator]

@@InventiveHarvestA set with no elements is the same as the empty set. So you can say that 0 is a set with no elements or say the same thing by saying that zero is the empty set.

[@lillymariscal6785]

P R O M O S M 😈