It is unlikely we'd see something that simply is not there. Here's the first question again, in the hopes you'll read and comprehend it this time:
Where in Cantor's Theorem (or in the proof to Cantor's Theorem) is this construction method? Don't keep giving us what you imagine to be an example of its use. Point to the construction method directly.
That would be the post where you provided two separate mappings from S to P(S). The first was A->{A}; the second was A->{}.
Neither mapping is a bijection between the members of {A} and P({A}).
By way of vigorous hand-waving, you also generated a trivial mapping between {1,2} and {{},{A}}. This impressed nobody except you. Be that as it may, it is also not a bijective mapping between {A} and P({A}) or between the natural numbers and P({A}).
Care to try again?
The hand-waving is a direct result of you closed reasoning under X^2.
Your reasoning is a boring fantasy.
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Again (in this post a I correct some expressions):
jsfisher, it is not P(S) unless you account for every P(S) member.
Actually you can do it infinity many times and still you will not get a complete collection of different objects in both sides of the 1-to-1 mapping.
Be aware that "infinity many times" is a present continuous parallel reasoning's expression, which is not limited to your step-by-step serial only reasoning.
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By using a cross-contexts reasoning, let us use Cantor's construction method in order to define a 1-to-1 correspondence between natural numbers and any set of different objects.
First, let us demonstrate it on some finite case, for example, the powerset of set {1,2,3}.
P({1,2,3}) = ({},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
By using Cantor's construction method with the members of {1,2,3} and the same amount of members taken from P({1,2,3}), one enables to explicitly define each P({1,2,3}) member and put it with 1-to-1 correspondence with a natural number, such that there is no natural number that is not mapped with some P({1,2,3}) member.
Cantor's construction method constructs an explicit P({1,2,3}) member as follows:
1) The defined explicit P({1,2,3}) member is the result of a 1-to-1 correspondence between {1,2,3} members and the same amount of members taken from P({1,2,3}).
2) The explicit constructed P({1,2,3}) member is the result of some 1-to-1 correspondence, such that it includes a {1,2,3} member only if this member does not exist as one of the members of the P({1,2,3}) member that is in a 1-to-1 correspondence with it.
By using this construction method several times, one enables to define a 1-to-1 correspondence between natural numbers and P({1,2,3}) members, as follows:
Round 1 particular case:
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1 ↔ {1} |
2 ↔ {2} | Provides
1 ↔ {}
3 ↔ {3} |
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Round 2 particular case:
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1 ↔ { } |
2 ↔ {2} | Provides
2 ↔ {1}
3 ↔ {3} |
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Round 3 particular case:
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1 ↔ {1} |
2 ↔ { } | Provides
3 ↔ {2}
3 ↔ {3} |
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Round 4 particular case:
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1 ↔ {1} |
2 ↔ {2} | Provides
4 ↔ {3}
3 ↔ { } |
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Round 5 particular case:
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1 ↔ { } |
2 ↔ {3} | Provides
5 ↔ {1,2}
3 ↔ {1,3} |
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Round 6 particular case:
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1 ↔ { } |
2 ↔ {2} | Provides
6 ↔ {1,3}
3 ↔ {1,2} |
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Round 7 particular case:
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1 ↔ {1} |
2 ↔ {3} | Provides
7 ↔ {2,3}
3 ↔ {1,2} |
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Round 8 particular case:
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1 ↔ { } |
2 ↔ {1} | Provides
8 ↔ {1,2,3}
3 ↔ {2} |
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The same 1-to-1 correspondence construction method works also in the case of infinitely many objects, where in this case {{}, ... , {1,2,3,...}} is the power set of {1,2,3,...}, and also in this case we are able to define a 1-to-1 correspondence between 1 to {}, 2 to {1,2,3,...}, and any natural number > 2 with any {{}, ... , {1,2,3,...}} member between {} and {1,2,3,...}.
It is done by using Cantor's construction method |{{}, ... , {1,2,3,...}}| times (again, it has to be taken in parallel, such that there is no process but simply a permanent incompleteness that can't be comprehended by using
only step-by step reasoning).
Since the set of
all powersets does not exist (because ...P(P({1,2,3}}}... size is unsatisfied) and since we are generally able (by using Cantor's construction method) to construct P(S) members by mapping S members with |S| P(S) members |P(S)| times, and also to define a 1-to-1 correspondence between the natural numbers and any P(S) degree, then also the size of the set of natural numbers is unsatisfied.
In other words, we discover that the concept of collection of different objects is essentially incomplete, or in other words, the whole is greater than the amount of any collection of different objects, where the whole is the cross-contexts existence among this infinity many context-dependent objects, which their amount is unsatisfied (they can't be summed into a whole).
