Cont: Deeper than primes - Continuation 1/3*9

Last edited:
doronshadmi said:
jsfisher, is it possible to define Cardinality only by <= relationship from set A to set B (written as A <= B)?

If no, then cardinality is a part of the definition of Cardinality as shown, for example, in http://www.internationalskeptics.com/forums/showpost.php?p=13144544&postcount=40 .

If you disagree with me, then please air your view about this post.

Thank you.
Click to expand...

Doron, jsfisher is not defining ‘cardinality’, he is defining ‘relative cardinality’, the relationship between the cardinalities of two sets.
 
doronshadmi said:
jsfisher, is it possible to define Cardinality only by <= relationship from set A to set B (written as A <= B)?
Click to expand...

As zooterkin properly points out, I did not define cardinality. I defined a relationship.

I chose "<=" as the symbol for this dyadic boolean relationship.

Cardinality of a set is normally indicated by vertical bars to either side of the set name -- a convention so well known to almost everyone that explicit mention of the convention should not be required.

In context, A and B would be interpreted as symbols for arbitrary sets -- another convention for which no explicit mention should be necessary.

To define the relationship, it must be presented with operands so the operands can participate in the definition.

Ergo, "|A| <= |B|" is the thing being defined. I.e., "the relationship symbolized by <= with operands, |A| and |B|".

And "there exists an injection from A to B" is the definition.

So, once again, cardinality is not being defined, but a relationship between the cardinality of two sets is being defined.


This is not third grade math, so please stop trying to apply third grade concepts to it.
 
jsfisher said:
As zooterkin properly points out, I did not define cardinality. I defined a relationship.
Click to expand...
Sorry jsfisher, here is what you have said:
jsfisher said:
For that matter, you need to be clear what you mean by "cardinality". No, wait, better still, I will provide you a definition of cardinality for the purposes of this discussion.

Cardinality is a relative measure of "size" of sets where |A| <= |B| if and only if there exists an injection from A to B. (The meanings for strict equality and strict inequality of cardinalities follow directly.)​

Note that this definition requires only the introduction of mappings into the set theory.
Click to expand...
No mention of a definition of Relative cardinality, isn't it?

So, after some number of posts and a little help from zooterkin, your definition of Cardinality, is not a definition of Cardinality, but it is actually a definition of a relative measure between the cardinalities of two arbitrary sets.
In other words, your determination of the need to be clear is not fulfilled.

---------------------

Any way, 'The definition of a relative measure between the cardinalities of two arbitrary sets' has a motivation behind it, which is:

To be general enough, in order to not distinguish between finite or infinite sets.

ZF(C) is actually a theory of sets, where its Axiom Of Infinity, is not actually Axiom Of Infinity unless more things are involved with it.

This style of work is done for the name of generalization, such that any given statement does not stand for its own, but depends on the interpretation that is given to it according to the rules of a given context.

In other words, traditional mathematics is based on the of Philosophy of relativism (https://plato.stanford.edu/entries/relativism/) and jsfisher actually demonstrates it, no matter what he argues about the independence of traditional mathematics in any philosophical issues.
 
Last edited:
Now back to the current issue, my answer to http://www.internationalskeptics.com/forums/showpost.php?p=13126898&postcount=3487 is given in http://www.internationalskeptics.com/forums/showpost.php?p=13142488&postcount=31 .

According to a reasonable criticism (that unlike you, is aware of the philosophical point of view of it) it will be changed.

If you still reject Philosophy as a source of motivation for mathematical work, any mathematical work, then there is no real discussion between us, because all you care is to show how Philosophy is disconnected from actual mathematical work (which is by itself, no more no less than some philosophical point of view).

Your particular philosophical view of mathematics is actually like a string that penetrates trough all your posts since our first dissuasion that was done almost 12 years ago.

I have learned a lot from your particular philosophical view of mathematics, and tried to air my particular philosophical view of mathematics, which unlike your philosophical point of view, does not reject philosophy as an actual factor of mathematical work, any mathematical work.
 
Last edited:
doronshadmi said:
If you still reject Philosophy as a source of motivation for mathematical work, any mathematical work, then there is no real discussion between us, because all you care is to show how Philosophy is disconnected from actual mathematical work (which is by itself, no more no less than some philosophical point of view).
Click to expand...


You are just now figuring out that the whole point of building Mathematics on a axiomatic basis is to be disconnected, as you say, from Philosophy? You need to pay more attention.

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.
 
jsfisher said:
... but it does not have the consistency nor any sort of rigor essential for advancing Mathematics.
Click to expand...
This is no less no more than your philosophical point of view about the relations between Mathematics and Philosophy.

This approach of separated islands of context dependent mathematical frameworks, is a direct result of the understanding that most of these separated and, so called, rigor contexts dependent islands (which are able to deal with Arithmetic) can't prove their own consistency by finitely many axioms (the end of Hilbert's program, as a result of Godel's incompleteness theorems).

The end of Hilbert's program, is one of the reasons why mathematicians stopped dealing with "big ideas" (philosophical notions) and reduced their work to "small ideas" (the technical mechanism of rigorous methods).

----------------------

Please critisize http://www.internationalskeptics.com/forums/showpost.php?p=13142488&postcount=31 and it's links, by your, so called, rigorous mathematical working methods.
 
Last edited:
jsfisher said:
You are just now figuring out that the whole point of building Mathematics on a axiomatic basis is to be disconnected, as you say, from Philosophy? You need to pay more attention.
Click to expand...
You need to pay more attention of the fact that most of the interesting mathematical frameworks that are based on axioms, can't prove their own consistency, so you are working most of the time in a very shallow water, exactly because of what you call rigor.
 
Last edited:
Little 10 Toes said:
Please provide evidence of this claim.
Click to expand...

He's right about that consistency thing. Gödel’s second Incompleteness Theorem proved that any consistent formal system in Mathematics rich enough to include arithmetic cannot prove its own consistency.

Be that as it may, though, the point is irrelevant in the current discussion arc. Mathematics shuns inconsistency even though it cannot usually guarantee its absence while Philosophy cherishes it.
 
jsfisher said:
He's right about that consistency thing. Gödel’s second Incompleteness Theorem proved that any consistent formal system in Mathematics rich enough to include arithmetic cannot prove its own consistency.

Be that as it may, though, the point is irrelevant in the current discussion arc. Mathematics shuns inconsistency even though it cannot usually guarantee its absence while Philosophy cherishes it.
Click to expand...
It is relevant, as shown in http://www.internationalskeptics.com/forums/showpost.php?p=13147204&postcount=52 , and my argument is not about Mathematics vs. Philosophy, but it is about the philosophical motivations that actually deeply influence on the rigorous mechanism of any mathematical work.

As for the current discussion Zermelo and his followers vary carefully established a theory of sets such that it will preserve Cantor's notions of infinite sets in terms of actual infinity.

For example, Cardinality is not defined directly by observations of natural numbers (as done for example in http://www.internationalskeptics.com/forums/showpost.php?p=13142488&postcount=31, and was improved by jsfisher's criticism) , but it is defined indirectly through a relative measurement (a one-to-one mapping in this case) from the members of one set (called A) to the members of another set (called B), in such a way that enables to plug in Cantor's transfinite system (which is actually the translation of a philosophical notion of infinity in terms of a set that actually has infinitely many members that are taken as a complete whole).

By this indirect definition of Cardinality one enables to suddenly jump into a measurement unit like aleph-0, like a third degree magician, which truly believes that that by indirect definition of Cardinality the audience will not pay attention to his inconsistent trick.

Unlike jsfisher indirect approach, which also ignore the philosophical motivations behind his indirect definition of cardinality, http://www.internationalskeptics.com/forums/showpost.php?p=13142488&postcount=31 does not need any indirect methods in order to define Cardinality, and I do not try to hide my philosophical motivations, which argue that any infinite collection (where an infinite set is a particular case of an infinite collection) is no more than potentially infinite since its accurate size is by definition not satisfied.

By going beyond the notion of collections, and by using philosophical direct approach of the discussed subject, actual infinity is essentially non-composed, as already demonstrated in http://www.internationalskeptics.com/forums/showpost.php?p=12584400&postcount=3285 .
 
Last edited:
jsfisher said:
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.
Click to expand...
Let's move on jsfisher.

So according to traditional mathematics:

1) Cardinality is a relative measure of "size" of sets where |A| <= |B| if and only if there exists an injection from A to B.

2) The ZF(C) axiom of infinity has two properties:

Property 1: ∅ ∈ I

Proeprty 2: ∀ x ∈ I ( ( x ∪ { x } ) ∈ I )


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)).
 
Last edited:
Little 10 Toes said:
Why? If you read the Axion of Infinity, it already postulated that the set I is infinite.
Click to expand...

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.
 
You must log in or register to reply here.

Share this page

ISF - Join now!

Every member here is approved by hand. No bots, no spam, just people who care about evidence and honest debate.

Membership is free!

Create your free account

Back
Top Bottom