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Question regarding Infinity
- Thread starter GLHF22
- Start date
- 50
- 5
Normally when people write 2^X for cardinal numbers they're referring to cardinal exponentiation, which is defined by the Power Set. For finite numbers the Power Set equals ordinal exponentiation, but when it comes to transfinite numbers it doesn't.
In math we have something known as hyper operators.
H0(a,b) = a + 1
H1(a,b) = a + b
H2(a,b) = a + a + a + ... (b times) = a*b
H3(a,b) = a*a*a*... (b times) = a^b
H4(a,b) = a^a^a^... (b times) = aÔåæb
...etc.
No hyper operation is going to make your infinity larger. That's why you have ordinals like epsilon, which is defined as w^w^w^... (where w is an transfinite ordinal) but is still just countable.
But the Power Set is defined completely differently, it's a function of combinatorics. That's why most teachers use P(X) or ^X 2 (where the superscript comes before the 2) for the Power Set, to not confuse their students that they're dealing with regular exponentiation.
In math we have something known as hyper operators.
H0(a,b) = a + 1
H1(a,b) = a + b
H2(a,b) = a + a + a + ... (b times) = a*b
H3(a,b) = a*a*a*... (b times) = a^b
H4(a,b) = a^a^a^... (b times) = aÔåæb
...etc.
No hyper operation is going to make your infinity larger. That's why you have ordinals like epsilon, which is defined as w^w^w^... (where w is an transfinite ordinal) but is still just countable.
But the Power Set is defined completely differently, it's a function of combinatorics. That's why most teachers use P(X) or ^X 2 (where the superscript comes before the 2) for the Power Set, to not confuse their students that they're dealing with regular exponentiation.
- 50
- 5
Yes.GLHF22 said:
X^w (where X is a cardinal and w is an ordinal) isn't really correct in the first place because you're mixing two incompatable classes of numbers.GLHF22 said:
And X^X doesn't make any sense, because while 2^X is defined as the Power Set, there is no definition for 3^X or 7^X or X^X.
That said, ordinals are used for the subscripts of cardinals. X_0, X_1, X_2, ..., X_w, X_(w+1).
While w^w and w are equal in size ( card(w^w) = card(w) = X ) the well-ordering property makes it so that w^w isn't equal to w. But the well-ordering has nothing to do with size, 1+w = w Ôëá w+1 (consequence of the well ordering property).
The first uncountable ordinal (aleph-1) is the union of all countable ordinals (and not a hyper-operator function of any kind).
