Actual infinity, if I got it right, consists of considering the set of natural numbers as an entity in itself. In other words 1, 2, 3,.. is a potential infinity but {1,2, 3,...} is an actual infinity. In symbolic terms it seems the difference between them is just the presence/absence of the curly braces, } and {. — TheMadFool
Axiom of infinity. That's as subtle as a gun in your face I guess. I don't know. Am I making sense here? — TheMadFool
In symbolic terms it seems the difference between them is just the presence/absence of the curly braces, } and {. — TheMadFool
In other words there is a largest number and infinity doesn't exist. — TheMadFool
If l say, l have taken 100 steps, l am using the word "steps" in a different sense, to mean a numerical quantity.to take an infinite number of steps
In other words there is a largest number and infinity doesn't exist. — TheMadFool
intensional definition gives the meaning of a term by specifying necessary and sufficient conditions for when the term should be used.
This is the opposite approach to the extensional definition, which defines by listing everything that falls under that definition
I'm confused by the distinction actual vs potential infinity? — TheMadFool
Axiom of infinity. That's as subtle as a gun in your face I guess. I don't know. Am I making sense here? — TheMadFool
Looks like the difference between Platonism and Constructivism. — Marchesk
it should be #{1,2, 3,...} or card({1,2, 3,...}) or |{1,2, 3,...}| for actual infinity — alcontali
It is very difficult to define infinity using any concept other than infinity itself. Hence it is often circular, self referential. — Wittgenstein
Without the axiom of infinity, a concept of actual infinity is not viable. — alcontali
The set of natural numbers does not have an upper bound, so it will always have a number that is smaller than another number. — Wittgenstein
The problem with axiom of infinity is that it fails to fall in one of the two categories. Intension and extension.
"intensional definition gives the meaning of a term by specifying necessary and sufficient conditions for when the term should be used."
"This is the opposite approach to the extensional definition, which defines by listing everything that falls under that definition." — Wittgenstein
Some logician view that infinite extensions are meaningless as extensions must be complete in order to be well defined, so infinity cannot be defined by extensions. ( They reject Cantors proof too )
The problem with definition using intention is that they are circular. — Wittgenstein
in Peano arithmetic, the collection of all the natural numbers is a proper class. — fishfry
Everything is a set, including tuples, sequences, and multisets. — GrandMinnow
There is an "isomorphism" between tuples and finite sequences. For example: The tuple <x y z> "encodes the same information" as the sequence {<0 x> <1 y> <2 z>}. — GrandMinnow
Your claimed proof that there is no infinite set is not recognizable as a proper mathematical argument but instead proceeds by hand waving non sequitur — GrandMinnow
(1) What math book is that? What is the context? What does the variable 'u' range over? What specific operation does '+' stand for? — GrandMinnow
2) There is no mathematical object named 'infinity — GrandMinnow
(3) Your "nothing = infinity" is just wordplay. — GrandMinnow
From what axioms, definitions, and rules of inference do you argue that? — GrandMinnow
I wrote "isomorphism" in scare quotes because I don't mean an actual function. I mean that tuples and sequences are "isomorphic" in that you can recover the order from one to the other and vice versa. This can be expressed exactly, but it's a lot of notation to put into posts such as these. Anyway, the general idea is obvious and used in mathematics extensively. — GrandMinnow
A very simple text. — TheMadFool
Axiom of infinity? — TheMadFool
How is it "wordplay"? — TheMadFool
Oh I see now. They may not be the same thing but just two different objects that behave in the same way. — TheMadFool
what axioms would be necessary for the existence of natural numbers and the basic mathematical operations of + and ×? I begin from these — TheMadFool
This can be expressed exactly, but it's a lot of notation to put into posts such as these. — GrandMinnow
\[\n((\d*(\.\d*)? ?)*\n?)*\]
The following set of sets is an element of the powerset of real numbers: — alcontali
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