No, cardinality, as a relative measure of set size, can be defined by defining a relationship between the sizes of two sets. Id est, |A| <= |B| if and only if there exists an injection from A to B.
2) The ZF(C) axiom of infinity has two properties:
No, the Axiom establishes the existence of (at least one) set, I, that has those two properties.
Please (by using (1),(2) and (if needed) more ZF(C) axioms) rigorously establish I as an infinite set (which means that its cardinality is strictly greater than any natural number (where 0 is also taken as a natural number)).
You are the one trying to discredit "traditional mathematics." The burden to prove or disprove whatever it is that establishes your claim lies with you.
No, cardinality, as a relative measure of set size, can be defined by defining a relationship between the sizes of two sets. Id est, |A| <= |B| if and only if there exists an injection from A to B.
Actually, it doesn't. The Axiom establishes that there exists a set meeting two properties. You need to define what it means for a set to be infinite before you can establish that that set meets the definition.
The normal approach to that is to adopt a measure of set size, and then it follows.
Philosophy credits itself with having "really good ideas"TM, which it may or may not have, but it does not have the consistency nor any sort of rigor essential for advancing Mathematics. The Dark Ages are over; time you moved on.
You are the one with something to prove (or disprove), not me. These attempts of yours to reverse responsibilities or divert to side topics do not advance your cause.
It would be especially good for you to start with a statement of just what is it you are claiming -- a nice, simple declaration of your treatise. Is that too much to ask?
You are the one with something to prove (or disprove), not me. These attempts of yours to reverse responsibilities or divert to side topics do not advance your cause.
It would be especially good for you to start with a statement of just what is it you are claiming -- a nice, simple declaration of your treatise. Is that too much to ask?
It would be especially good for you to start with a statement of just what is it you are claiming -- a nice, simple declaration of your treatise. Is that too much to ask?
It would be especially good for you to start with a statement of just what is it you are claiming -- a nice, simple declaration of your treatise. Is that too much to ask?
It would be especially good for you to start with a statement of just what is it you are claiming -- a nice, simple declaration of your treatise. Is that too much to ask?
Yes, it is. I have nothing by itself to prove in this thread, Doronshadmi. You do. You started the thread, after all, presumable to make some point.
It would be especially good for you to start with a statement of just what is it you are claiming -- a nice, simple declaration of your treatise. Is that too much to ask?
Until this very moment you did not address those quotes consistently and rigorously, which is, as you say, "essential for advancing Mathematics".
My deceleration is this:
Nothing is normal (where in this context normal means: consistently and rigorously) of how traditional mathematics "adopt a measure of set size" in terms of the non-finite, so nothing follows because it is arbitrarily done by it.
If you disagree with my deceleration, please consistently and rigorously support your quotes above.
No one will take away from you the freedom to evade my deceleration.
jsfisher you, of course, free to do whatever you like.
Until this very moment you did not address those quotes consistently and rigorously, which is, as you say, "essential for advancing Mathematics".
My deceleration is this:
Nothing is normal (where in this context normal means: consistently and rigorously) of how traditional mathematics "adopt a measure of set size" in terms of the non-finite, so nothing follows because it is arbitrarily done by it.
If you disagree with my deceleration, please consistently and rigorously support your quotes above.
No one will take away from you the freedom to evade my deceleration.
You have a claim. You provide evidence that you are right.
You have not provided evidence that you/your belief system is/are right.
And you are not accurately quoting jsfisher. His full quote is
Philosophy credits itself with having "really good ideas"TM, which it may or may not have, but it does not have the consistency nor any sort of rigor essential for advancing Mathematics. The Dark Ages are over; time you moved on.
His quote, simplistically, is addressing that Philosophy does not advance Mathematics. I, generally speaking with my basic understanding of philosophy, agree with that.
If you disagree with jsfisher's claim, challenge him on it and ask for proof/evidence. If you want, you can provide evidence that philosophy DOES have consistency and rigor essential for advancing Mathematics.
Just to be clear doronshadmi, your issue is about how mathematics defines cardinality? Nothing else, just how cardinality is defined? A simple Yes/No answer is fine.
Edit: Once your viewpoint is clear, it can be addressed.
So? The out-of-context quotation isn't an attempt to advance Mathematics.
If you wish to challenge my statement, then feel free to do so, but include some basis for your objection that can become the basis for discussion (if anyone cares to engage).
My deceleration is this:
Nothing is normal (where in this context normal means: consistently and rigorously) of how traditional mathematics "adopt a measure of set size" in terms of the non-finite, so nothing follows because it is arbitrarily done by it.
Your treatise is difficult to parse. Do not introduce non-standard usage of a word like 'normal' if you mean something else. Use those something-else words. Your bad habit of continually adding unnecessary words only muddles you posts.
It appears you are suggesting that there is something wrong with Mathematics for having ways to measure the size of a set. Is that right? You would prefer cardinality not be defined?
I claim that the measure of the size of a non-finite set, is not done consistently and rigorously (which means that it is done arbitrarily) by traditional mathematics.
So, cardinality is not being defined by traditional mathematics.
In that case, why do you asking me if I "would prefer cardinality not be defined?" when you are explicitly state that (by traditional mathematics) "cardinality is not being defined"?
As for me, I prefer cardinality to be defined without eliminating the difference between finite and non-finite cardinality, for example:
1) I do not need iff in order to define Cardinality by the members of the set of natural numbers (notated as N), where natural numbers (including number 0) are naturally understood (no extra maneuvers are needed).
2) I provide symbols to the concepts "less than", "equal to", "greater than" (which I tend to replace by "more than") which are naturally understood (no extra maneuvers are needed).
3) I use iff in a vary simple way in case of relative measure, between two arbitrary sets A and B, where their cardinalities (what is compared) and the terms of how they are compared are simply and intuitively addressed, because of the simple use of iff.
4) By being simple and intuitive in definitions 1 and 2, I am able to very simply define finite (definition 3) and non-finite (definition 4) sets, without any need to add anything, which are not already given by definitions 1 to 4.
5) Definition 5 very simply addresses non-strict inequality as a range between strict inequality (< or >) and equality (=).
Just to be clear doronshadmi, your issue is about how mathematics defines cardinality? Nothing else, just how cardinality is defined? A simple Yes/No answer is fine.
Edit: Once your viewpoint is clear, it can be addressed.
I claim that the measure of the size of a non-finite set, is not done consistently and rigorously (which means that it is done arbitrarily) by traditional mathematics.
What an odd thing to say. For one, arbitrariness has nothing to do with rigor and consistency. For another, where is the inconsistency in defining a relationship by "|A| <= |B| if and only if there is an injection from A to B"? Where is the lack of rigor?
Subtlety and nuance escape you. I did not define cardinality, at least not directly. Euclidean Geometry does not define point (directly), nor does ZF Set Theory define set (directly).
Instead, ZF Set Theory provides a set of axioms and axiom schema that characterize set properties; Euclidean Geometry characterizes points by way of its postulates; and I defined a relationship sufficient for comparing cardinalities of sets.
Instead, ZF Set Theory provides a set of axioms and axiom schema that characterize set properties; ... ; and I defined a relationship sufficient for comparing cardinalities of sets.
Once again you are not consistently and rigorously demonstrate how ZF and your "relationship sufficient for comparing cardinalities of sets" establish a set with non-finite cardinality.
Once again you are not consistently and rigorously demonstrate how ZF and your "relationship sufficient for comparing cardinalities of sets" establish a set with non-finite cardinality.
You need to define what it means for a set to be infinite before you can establish that that set meets the definition.
The normal approach to that is to adopt a measure of set size, and then it follows.
The highlighted quotes are no more than philosophical declarations, and there is here in this thread a person that says that he can address those declarations consistently and rigorously, which is "essential for advancing Mathematics".
Until this very moment this person did not consistently and rigorously established a non-finite set, and in order to avoid it, he tries to represent the discussion in terms of some kind of personal discussion, as clearly seen in the first quote of this post.
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