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Maybe somebody should add a note to the article explaining that the "subset" referred to is the notion of subset within the theory (in an attempt to ward off the confusion you typically get when somebody first hears that ZFC has countable models). -- Cwitty 01:11, 22 November 2003
isn't it possible to replace (∀ D, D ∈ C → D ∈ A) by C ⊂ A which I would find easier to understand.
The article states that the Cartesian product of X and Y is a subset of the power set of the power set of the union of X and Y. But isn't it a subset of the power set of the union of X and Y? Why the double power set? Am I missing something?
X = {a, b} Y = {c, d}
XuY = {a, b, c, d}
P(XuY) = {{}, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, {a, b, c, d}}
XxY = {{a, c}, {a, d}, {b, c}, {b, d}}
I probably am. But what is it, I wonder... 24.238.113.229 ( talk) 05:52, 8 June 2008 (UTC)
I mean, what about "the set of all sets". Surely "the set of all sets" cannot have a power set as the power set of "the set of all sets" would be a set larger then "the set of all sets". But "the set of all sets" is the set of all sets and therefore the largest set. So this axiom leads too absurdity. I for one would rather ditch the axiom of power sets then the axiom of self consistency, but that is just me. It's "All that is is The All and yet The All is All that is." Vs. "Well surely there is something more then The All which contains The All. That which is above is as below, ya know." I mean it really cuts deep into the hermetic controversy. --Michaelidman —Preceding unsigned comment added by 66.31.206.34 ( talk) 04:15, 5 January 2009 (UTC)
There are multiple errors due to not dealing with things in precise mathematical form. First mathematicians erroneously believe the powerset of an infinite set is larger, to be consistent with ∞+2=the same ∞ it is not, all infinite sets of set theory type are the same size, neverending, which is not a number. The set of all sets is a complex concept and can be any of multiple formulations. Saying 'of all sets' is merely all finite sets, saying 'the set of …' changes the definition of 'set' which again changes the definition of set again etc. Referring to 'the set of all sets that contain themselves' further requires mathematicaly precise definition. Doing this in English reduces this to phylosophy. So they decided 'of all sets' is a class and powerset of an infinite set exponentiates it's size, not realizing powerset of an infinite set is still not precisely defined. Victor Kosko ( talk) 00:24, 12 January 2023 (UTC)
In this article, the inductive definition of the cartesian product is not associative. So, Ax(BxC)≠(AxB)xC. -- Gallusgallus ( talk) 18:54, 29 March 2010 (UTC)
The definition of the power-set axiom here has this, and I have seen this even in other texts. What would be the benefit of using y instead? PicoMath ( talk) 18:47, 9 November 2023 (UTC)
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Maybe somebody should add a note to the article explaining that the "subset" referred to is the notion of subset within the theory (in an attempt to ward off the confusion you typically get when somebody first hears that ZFC has countable models). -- Cwitty 01:11, 22 November 2003
isn't it possible to replace (∀ D, D ∈ C → D ∈ A) by C ⊂ A which I would find easier to understand.
The article states that the Cartesian product of X and Y is a subset of the power set of the power set of the union of X and Y. But isn't it a subset of the power set of the union of X and Y? Why the double power set? Am I missing something?
X = {a, b} Y = {c, d}
XuY = {a, b, c, d}
P(XuY) = {{}, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, {a, b, c, d}}
XxY = {{a, c}, {a, d}, {b, c}, {b, d}}
I probably am. But what is it, I wonder... 24.238.113.229 ( talk) 05:52, 8 June 2008 (UTC)
I mean, what about "the set of all sets". Surely "the set of all sets" cannot have a power set as the power set of "the set of all sets" would be a set larger then "the set of all sets". But "the set of all sets" is the set of all sets and therefore the largest set. So this axiom leads too absurdity. I for one would rather ditch the axiom of power sets then the axiom of self consistency, but that is just me. It's "All that is is The All and yet The All is All that is." Vs. "Well surely there is something more then The All which contains The All. That which is above is as below, ya know." I mean it really cuts deep into the hermetic controversy. --Michaelidman —Preceding unsigned comment added by 66.31.206.34 ( talk) 04:15, 5 January 2009 (UTC)
There are multiple errors due to not dealing with things in precise mathematical form. First mathematicians erroneously believe the powerset of an infinite set is larger, to be consistent with ∞+2=the same ∞ it is not, all infinite sets of set theory type are the same size, neverending, which is not a number. The set of all sets is a complex concept and can be any of multiple formulations. Saying 'of all sets' is merely all finite sets, saying 'the set of …' changes the definition of 'set' which again changes the definition of set again etc. Referring to 'the set of all sets that contain themselves' further requires mathematicaly precise definition. Doing this in English reduces this to phylosophy. So they decided 'of all sets' is a class and powerset of an infinite set exponentiates it's size, not realizing powerset of an infinite set is still not precisely defined. Victor Kosko ( talk) 00:24, 12 January 2023 (UTC)
In this article, the inductive definition of the cartesian product is not associative. So, Ax(BxC)≠(AxB)xC. -- Gallusgallus ( talk) 18:54, 29 March 2010 (UTC)
The definition of the power-set axiom here has this, and I have seen this even in other texts. What would be the benefit of using y instead? PicoMath ( talk) 18:47, 9 November 2023 (UTC)