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/sci/ - Science & Math

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16025764 No.16025764 [Reply] [Original]

can someone explain to me how the power set leads to a greater cardinality than aleph null? what stops the natural numbers from being enough to do a bijection? can't you just do a one to one correspondence via mathematical induction? and how does it prevent a cardinality existing that's between the integers and real numbers? is this all due to power sets?

>> No.16025766

basically, is the invocation of transfinite recursion done exclusively with power sets?

>> No.16025793

homework thread

>> No.16025795

>>16025793
this isn't homework i'm just trying to start a discussion on the topic of large cardinals.

>> No.16025812

>>16025764
Suppose X is any set and we have a surjection f: X -> P(X). Using specification, define the set S = {x in X | x not in f(x)}. Since f is surjective, we have y in X with f(y) = S. However, examining the situation closely we see that y in S <=> y not in S, which is a contradiction. So, there can never be a surjection from a set to its own power set.

>> No.16025813
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16025813

>>16025764
Large cardinals do not exist so thinking about them is a waste of time

>> No.16025815

>>16025764
With regards to cardinalities between |N| and |R|, the power set operation does not preclude any from existing. In fact, the question of whether they do is the continuum hypothesis which is famously independent of ZFC. So it proves |N| < |P(N)| = |R| but it cannot prove whether there are other cardinalities between those.

>> No.16026468

>>16025815
so is the cardinality of |R| 2^aleph zero, or just aleph one? im so confused

>> No.16026494

>>16026468
it is 2^(aleph_0). the question of whether this is also equal to aleph_1 is known as the Continuum Hypothesis (more accurately: the hypothesis is that the are equal)

>> No.16026540

>>16026468
https://youtu.be/SrU9YDoXE88?t=17m

>> No.16026551

>>16026540
https://youtu.be/SrU9YDoXE88?t=23m10s

>> No.16026579
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16026579

>Measurable cardinal
>can't measure it.

>> No.16026795

>>16025764
real (actually fake) numbers are a religion. the set of "computable" i.e. the actual existing numbers is countable according to the priests of set theory.
so we have absolutely no way of creating, seeing or knowing anything about those "reals" (fakes). just like religion there is no evidence except some vague notion of a very small epsilons and deltas which themselves actually can't be shown to exist