And I don't understand the rest of your post, starting with "any language expression that matches only this kind of stuff, would be the membership function for a set of which the cardinality would be the powerset of real numbers" — GrandMinnow
Whether it's "true" in any meaningful sense is, frankly, doubtful. — fishfry
No, you just made the same mistake I pointed out the first time. — GrandMinnow
but I don't have much problem understanding that the set of natural numbers and other infinite sets exist as abstract mathematical objects — GrandMinnow
And card(PPR) = beth2. — GrandMinnow
\[\n((\d*(\.\d*)? ?)*\n?)*\]
card("\[\n((\d*(\.\d*)? ?)*\n?)*\]")=beth2
It is not necessary to adopt platonism to accept that there are infinite sets. One may regard infinite sets as abstract mathematically objects, while one does not claim that abstract mathematical objects exist independently of consciousness of them. — GrandMinnow
There is an inherent contradiction in asserting that a symbol like 2 signifies an object, because the unifying agent which makes 2 into one object has not been identified, therefore that two are one object has not been justified, and there really is no such object. — Metaphysician Undercover
When we request justification, we see that "infinite set" is contradictory, as are most mathematical objects. — Metaphysician Undercover
The concept of infinite set is abstract and very Platonic but not contradictory. — alcontali
It is contradictory, because a set is closed, complete, (as an object it is bounded, defined) whereas an infinity of anything is open, incomplete, unbounded and indefinite. — Metaphysician Undercover
If they are a "pair" of shoes, this does not make them into an object, it is just another way of saying that they are two — Metaphysician Undercover
And if I cut a cat in two, there are two pieces of one cat. — unenlightened
Yes, but you are not allowed to put physical objects into a mathematical set.
You can only fill it up with language expressions.
So, if you cut a "cat" in two, you get {"c", "at"} or {"ca","t"}. — alcontali
This is more a philosophical or psychological question than a purely mathematical one, but I don't have much problem understanding that the set of natural numbers and other infinite sets exist as abstract mathematical objects. — GrandMinnow
I wouldn't state it that way. If we mean first order Peano arithmetic (PA), then there are not in PA definitions of 'set', 'class', and 'proper class'. Meanwhile, in set theory, the domain of the standard model of PA is a set.
19 hours ago — GrandMinnow
Your idealism solves the problem of contradiction but at the price of failing to account for how we actually talk about the world. — unenlightened
All possible sentences you can say in English is a set. — alcontali
But what do i have to do to make them one, tie the laces together - glue the soles together - crush them into a singularity? — unenlightened
So don't do it. — unenlightened
No it isn't. A set consists of objects, not possible objects. — Metaphysician Undercover
S1 and S2 describe the same set. Therefore, S1 = S2. — alcontali
That two things are equal does not mean that they are the same. This is a known deficiency of mathematics, equality cannot replicate identity. Anyone who argues that 2+2 is the same as 4 needs to learn the law of identity, and respect the difference between equality and identity. — Metaphysician Undercover
The sentence "they both describe the same set and therefore they are extensional" is therefore in accordance with the axiomatic foundation of ZFC set theory. — alcontali
ZFC theory allows that two distinct things are the same, contrary to the law of identity. — Metaphysician Undercover
though they are "the same set" according to the deficient standard of ZFC set theory — Metaphysician Undercover
ZFC was initiated by Cantor and Dedekind in the 1870s — alcontali
LOL. Impressive Wiki skills. No bearing on the topic at hand. What can I say? — fishfry
These issues are thoroughly discussed in a nice paper by Barry Mazur, "When is one thing equal to some other thing?"
http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf — fishfry
The axiom of extensionality depends on the law of identity, which is a principle of logic and not of set theory. A thing is equal to itself. Then we define two sets to be equal if they have "the same" elements, meaning that we can pair off their respective elements using the law of identity. — fishfry
I am concerned with the principles of the system, not any installed base, or legacy, these are irrelevant to the acceptability of the principles. I know that you believe axioms are completely arbitrary, making such things very relevant, so join the mob, if you like the "mob rules" philosophy. — Metaphysician Undercover
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