What is aleph-null?
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What is aleph-null?
What is aleph-null?
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aleph null denotes the cardinality of the natural numbers, i.e. the total number there are of integers
it is the smallest infinite number and smaller than the car finality of the real numbers
Meaningless babble. Infinite sets do not have sizes.
that's why i said "cardinality" instead
The notion of size assigned to the smallest possible set provided by the axiom of infinity in Zermelo-Fraenkel set theory.
Size is an idea and can be interpreted however you like. Area is a type of size. There are infinitely many points in a square. Don't squares have area? Math is about building around formal thinking, not rejecting ideas because you think they're stinky.
You used 'i.e.' incorrectly. 'I.e' means 'that is.' Also it isn't number. Numbers are sets with well defined arithmetic operations attached.
The existence of such a notion of size follows directly from axioms, there is no delusion. You may disagree with fundamental mathematical axioms, but that doesn't make their consequences wrong. It is important to note that if you personally disagree with the existence of aleph-null, you would also disagree that the number line you learned as a child exists.
>if you personally disagree with the existence of aleph-null, you would also disagree that the number line you learned as a child exists
complete nonsense
k ill take the bait.
Refuting axiom of infinity but also asserting the existence of a naive number line implies claim implies there is a maximal integer. Please fix this inconsistency and I will yield to your big brain argument.
>existence of a naive number line implies claim implies there is a maximal integer.
Infinity means 'just keep getting bigger'. From this, it then extends to 'well, which infinity is bigger?'
>all math is set theory
countability doesn't come from the axiom of infinity.
>Numbers are sets with well defined arithmetic operations attached.
https://en.wikipedia.org/wiki/Cardinal_number#Cardinal_arithmetic
Areas have the same cardinality though so the finitist schizo would be correct in saying they have no "sizes" to compare
cool story. prove you aren't pseuding and define "cardinality" in your own words right now. specifically, prove to me you know what is meant by card(A)=card(B)
the delusion that infinity is countable
countable is a dumb word
denumerable / enumerable is better
🙂
The intersection of all inductive sets
It’s a israeli trick.
The zeroest zero
Let's just say they didn't pick a Hebrew letter for no reason.
Cantor's paradox shows that you cannot assign a cardinality to the set of all cardinal numbers. Couldn't this be resolved by defining a constant superinfinity with the unique property that it is so large, if you take the power set of a set with cardinality superinfinity then the power set has cardinality superinfinity as well? Either this or you have to accept that not all set have a well-defined cardinality.
In set theories with proper classes like NBG and MK, you can take the powerclass (class of all subsets of) a proper class.
How does this solve the problem? Do classes have no cardinality? If they do you could apply the same argument again.
>All sets have well-defined cardinality
Is this an axiom? Doesn't the Cantor paradox show the exact opposite?
>Russell's paradox
Not actually a paradox. It doesn't define a class either. Just shows how unrestricted comprehension can produce utter nonsense.
>Cantor's paradox shows that you cannot assign a cardinality to the set of all cardinal numbers.
no it doesnt
Dont stop him. Let him keep going.
All sets have well-defined cardinality. Your argument just shows that the premise that cardinal numbers are all included in a set is false, similar to how Russell's paradox doesn't prove that there exists sets who belong to themselves, but rather just proves that the Russell class is not a set.
What is Santa Claus? What is Easter Bunny? What is the Fairy Queen? The Boogeyman under your bed?