ZFC .... Axioms of Foundation .... and Infinity ....

[Math Amateur]

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume 1: Foundations and Elementary Real Analysis" ... ...

I am at present focused on Part 1: Prologue: The Foundations of Analysis ... Chapter 1: The Axioms of Set Theory ...

I need help with an aspect of the proof of Proposition 1.7.5 ...

Proposition 1.7.5 reads as follows:https://www.physicsforums.com/attachments/7003In the above proof we read the following:

"By the foundation axiom, there exists \(\displaystyle n \in \mathbb{Z}^+\) such that no member of \(\displaystyle f(n)\) is in \(\displaystyle f( \mathbb{Z}^+ )\). ... ... "

Can someone please explain how/why the foundation axiom implies that there exists \(\displaystyle n \in \mathbb{Z}^+\) such that no member of \(\displaystyle f(n)\) is in \(\displaystyle f( \mathbb{Z}^+ )\). ... ... ?

PeterNOTE:

To enable readers to follow the above post I am providing Garling's text on the foundation axiom and the axiom of infinity ... ...View attachment 7004
View attachment 7005

 

[Ackbach]

In the foundation axiom, set $f(Z^{+}) = A$. The foundation axiom asserts that there is some member $b\in f(Z^{+})$ such that $b\cap f(Z^{+}) = \varnothing$. But since $b\in f(Z^{+})$, it follows that there exists some $n\in Z^{+}$ such that $f(n)=b$. And there you have the assertion in Garling. Does that answer your question?

 

[Math Amateur]

In the foundation axiom, set $f(Z^{+}) = A$. The foundation axiom asserts that there is some member $b\in f(Z^{+})$ such that $b\cap f(Z^{+}) = \varnothing$. But since $b\in f(Z^{+})$, it follows that there exists some $n\in Z^{+}$ such that $f(n)=b$. And there you have the assertion in Garling. Does that answer your question?

Hi Ackbach ... sorry to be slow in answering... had to travel interstate ...

Thanks so much for the help ... very clear answer ... thanks

Peter

 

[Math Amateur]

Hi Ackbach ... sorry to be slow in answering... had to travel interstate ...

Thanks so much for the help ... very clear answer ... thanks

Peter

Hi again, Ackbach ...

Just a further question you may be able to help with ...

Just prior to Proposition 1.7.5 Garling writes:

"Let us use the foundation axiom to show that infinite regress is not allowed."

I do not fully understand how Proposition 1.7.5 demonstrates that infinite regress is not possible ... can you explain ...?

Peter

 

[Ackbach]

Hi again, Ackbach ...

Just a further question you may be able to help with ...

Just prior to Proposition 1.7.5 Garling writes:

"Let us use the foundation axiom to show that infinite regress is not allowed."

I do not fully understand how Proposition 1.7.5 demonstrates that infinite regress is not possible ... can you explain ...?

Peter

Hmm. Is there some text in Garling that's missing in your scan? If so, could you please supply that? I'm thinking in particular between the statement you quoted and Proposition 1.7.5.

 

[Math Amateur]

Hmm. Is there some text in Garling that's missing in your scan? If so, could you please supply that? I'm thinking in particular between the statement you quoted and Proposition 1.7.5.

Hi Ackbach,

The original post showed the complete text from the start of the foundation axiom through (and including) Proposition 1.7.5.

There is no missing text ...

Peter

 

[Ackbach]

Hi Ackbach,

The original post showed the complete text from the start of the foundation axiom through (and including) Proposition 1.7.5.

There is no missing text ...

Peter

I see. Well, what about after? Maybe the author's not done showing that infinite regress isn't allowed. Is there an indicator sentence anywhere later? Maybe after the next theorem? Or the one after that? I mean by "indicator sentence" something along the lines of, "And this shows that infinite regress is not allowed." In other words, I'm wondering if Theorem 1.7.5 is a lemma, and not the "final theorem" demonstrating that infinite regress is not allowed. I would agree with you that it's not obvious to me that Theorem 1.7.5 does the trick.

 

[Math Amateur]

I see. Well, what about after? Maybe the author's not done showing that infinite regress isn't allowed. Is there an indicator sentence anywhere later? Maybe after the next theorem? Or the one after that? I mean by "indicator sentence" something along the lines of, "And this shows that infinite regress is not allowed." In other words, I'm wondering if Theorem 1.7.5 is a lemma, and not the "final theorem" demonstrating that infinite regress is not allowed. I would agree with you that it's not obvious to me that Theorem 1.7.5 does the trick.

Thanks for that Ackbach ...

I have concluded that Garling (a bit like Rudin) can be very terse and cryptic...

But ... thanks again for your support and help ...

Peter

 

[Evgeny.Makarov]

Infinite regress would mean the existence of an infinite sequence $a_1\ni a_2\ni a_2\ni\dots$ of sets. Since $Z^+$ can be considered as the set of natural numbers (which is explained after Proposition 1.7.5), such sequence can be viewed as a function from $Z^+$, namely, $a_n=f(n)$, where $n$ in the right-hand side is $\emptyset$ with $n$ pluses. The proposition says that such sequence is impossible: there exists an $n\in Z^+$ for which $f(n^+)\notin f(n)$.

 

[Math Amateur]

Infinite regress would mean the existence of an infinite sequence $a_1\ni a_2\ni a_2\ni\dots$ of sets. Since $Z^+$ can be considered as the set of natural numbers (which is explained after Proposition 1.7.5), such sequence can be viewed as a function from $Z^+$, namely, $a_n=f(n)$, where $n$ in the right-hand side is $\emptyset$ with $n$ pluses. The proposition says that such sequence is impossible: there exists an $n\in Z^+$ for which $f(n^+)\notin f(n)$.

Thanks for the explanation/clarification Evgeny ...

I appreciate your help and support ...

Peter