1. I was reminded the other day of a conversation I had at the back-end of last year.

It seems there is a school of mathematical thinking whose slogan is: if you cannot give me explicit instructions on how to construct an object, I am under no obligation to accept its existence. This school is called constructivism.

Among other things this is to deny the Axiom of Choice, which is a part of the axiomatic set theory known as ZFC, and very widely regarded as the bed-rock of all modern mathematics. So let's see what this agenda might imply.

Off the top of my head, I would say that, by denying the AC, the well-ordering theorem - any set can be so ordered that each subset has a least element - need not apply in general; specifically, it may not apply to . Hurrah!

It may also deny the validity of the Banach-Tariski paradox - any n-ball can be "dissected" and "reassembled" in such a way as to generate two n-balls, each with the same volume as the first. Hurrah again!

It may also mean that there are vector spaces with no basis. Boo! to that.

And in general, the proofs of many well-known theorems may need to be re-written, and some of these theorems may actually be unprovable under the constructivist agenda. This would be a Bad Thing.

It is often frequently said that constructivists also deny the Law of Excluded Middle - a statement that is not true is of necessity false, no deals. I am not sure they all did, but then, I am asking for your thoughts on this

It seems that constructivists are a bit of a fringe. But before we dismiss it as a lunatic fringe, note that some extremely big hitters have, and do, adhere to it. I name "Fixed point" Brouwer and "delta" Kroenecker, for two.

I am also aware that, even now, there is a form of mathematical etiquette that requires any proof implicitly using AC to say so. This seems to imply a certain ambivalence to the axiom.

What think you all?

2.

3. Originally Posted by Guitarist
I was reminded the other day of a conversation I had at the back-end of last year.

It seems there is a school of mathematical thinking whose slogan is: if you cannot give me explicit instructions on how to construct an object, I am under no obligation to accept its existence. This school is called constructivism.

Among other things this is to deny the Axiom of Choice, which is a part of the axiomatic set theory known as ZFC, and very widely regarded as the bed-rock of all modern mathematics. So let's see what this agenda might imply.

Off the top of my head, I would say that, by denying the AC, the well-ordering theorem - any set can be so ordered that each subset has a least element - need not apply in general; specifically, it may not apply to . Hurrah!

It may also deny the validity of the Banach-Tariski paradox - any n-ball can be "dissected" and "reassembled" in such a way as to generate two n-balls, each with the same volume as the first. Hurrah again!

It may also mean that there are vector spaces with no basis. Boo! to that.

And in general, the proofs of many well-known theorems may need to be re-written, and some of these theorems may actually be unprovable under the constructivist agenda. This would be a Bad Thing.

It is often frequently said that constructivists also deny the Law of Excluded Middle - a statement that is not true is of necessity false, no deals. I am not sure they all did, but then, I am asking for your thoughts on this

It seems that constructivists are a bit of a fringe. But before we dismiss it as a lunatic fringe, note that some extremely big hitters have, and do, adhere to it. I name "Fixed point" Brouwer and "delta" Kroenecker, for two.

I am also aware that, even now, there is a form of mathematical etiquette that requires any proof implicitly using AC to say so. This seems to imply a certain ambivalence to the axiom.

What think you all?

If an investigation using the scientific method, to check the state of some singular unknown, returns the positive determination of its state, as being true. Then it cannot be false.

However, if the question allows multiple answers, or involves multiple fields of interest to be allowed into the question. Then it was not a real true false question. But rather a multiple choice question.

I believe what can take place, is the poor rules can become weapons. A poor question is a weapon.

An example. Are we going to allow the violent minority types the same knowledge as calm intelligent rational folk?

Now they have clearly made an accusation that there is some group of calm intelligent rational, nonviolent folk. I resent that statement. I know better.

I find the only way to get to truth, is to have a fun conversation with my friends about it. When I am having a great time laughing sharing, I realize some stupid thing I have locked away in my mind is incorrect. Or has my mind locked up.

If it gets too nasty. I need to do some homework before I can go back and discuss it with my friends. More then likely we are all good individuals, with misunderstandings and prejudices, basically we are ignorant. God did warn us of that. And shows us everyday that it is true, exponentially true.

As far as unprovable theorems, by definition. They are an assumption. A wild guess.

If stated like that, I would imagine that, they would be taken in context. And frankly, by the working masses avoided, and even ridiculed a bit.
Real close at hand, number crunching formulas and solutions, might serve man and God better.

If we found the caveman was etching, speed of light formula, on the wall. We would laugh because we know he died, at an early age, often of smoke inhalation. He did not control his food supply. So why would he be off with his head in the clouds with the speed of light formula?

Do not get me wrong, I like a sharpened well honed mathematical weapons chest, teetering on the cutting edge of reality and infinity. Ready to destroy the most complex problem.
But, the stuff off to the left, in the land of assumption, is not for me at this time, in our present state as humans. At this point we can only assume when our collapse will come. Not if it will come.

Kind of rambled on there. Sorry.

(Edited) I meant to start off, Great Post!

Sincerely,

William McCormick

4. Originally Posted by Guitarist
I was reminded the other day of a conversation I had at the back-end of last year.

It seems there is a school of mathematical thinking whose slogan is: if you cannot give me explicit instructions on how to construct an object, I am under no obligation to accept its existence. This school is called constructivism.

Among other things this is to deny the Axiom of Choice, which is a part of the axiomatic set theory known as ZFC, and very widely regarded as the bed-rock of all modern mathematics. So let's see what this agenda might imply.

Off the top of my head, I would say that, by denying the AC, the well-ordering theorem - any set can be so ordered that each subset has a least element - need not apply in general; specifically, it may not apply to . Hurrah!

It may also deny the validity of the Banach-Tariski paradox - any n-ball can be "dissected" and "reassembled" in such a way as to generate two n-balls, each with the same volume as the first. Hurrah again!

It may also mean that there are vector spaces with no basis. Boo! to that.

And in general, the proofs of many well-known theorems may need to be re-written, and some of these theorems may actually be unprovable under the constructivist agenda. This would be a Bad Thing.

It is often frequently said that constructivists also deny the Law of Excluded Middle - a statement that is not true is of necessity false, no deals. I am not sure they all did, but then, I am asking for your thoughts on this

It seems that constructivists are a bit of a fringe. But before we dismiss it as a lunatic fringe, note that some extremely big hitters have, and do, adhere to it. I name "Fixed point" Brouwer and "delta" Kroenecker, for two.

I am also aware that, even now, there is a form of mathematical etiquette that requires any proof implicitly using AC to say so. This seems to imply a certain ambivalence to the axiom.

What think you all?
In my experience the axiom of choice, in one form or another is used quite commonly and without much comment. There is no ambivalence about it and most mathematicians regard it as "obvious", which they must do since it is not provable. Paul Cohen showed years ago that the axiom of choice is independent of the usual Zermelo Frankel axioms. Zermelo Frankel plus choice is the usual standard for the foundations of mathematics.

There is a school of constructivists, Erret Bishop being the one most commonly mentioned, who have studied the degree to which much of modern mathematics can be developed without the axiom of choice. That is interesting work, but it is not widely used within the general mathematics community.

The bottom line is that there are many applications of the axiom of choice or an equivalent (Hausdorff maximality principle, well-ordering principle, Zorn's lemma, etc.) in the development of almost all of modern mathematics and those who do not accept it are decidedly out of the mainstream. Unless you are reading something that explicitly states that the axiom of choice has been excluded it is safe to assume that somewhere in the background it may have been used, unless you are dealing exclusively with finite sets or some very narrow subject in which you can verify that the axiom of choice is not needed.

5. The existence of non-measurable subsets of the real line is weaker then the axiom of choice (though stronger then countable choice, which most constructionists are happy with). I think it is equivalent to the boolean prime ideal thm, but I'm too lazy to check

In my experience, you only state your axioms (MA, CH, existence of certain large cardinals etc) if you are assuming anything past ZFC.

6. Originally Posted by river_rat
The existence of non-measurable subsets of the real line is weaker then the axiom of choice (though stronger then countable choice, which most constructionists are happy with). I think it is equivalent to the boolean prime ideal thm, but I'm too lazy to check

In my experience, you only state your axioms (MA, CH, existence of certain large cardinals etc) if you are assuming anything past ZFC.
I assume that CH is the continuum hypothesis.

What is MA ?

In my experience mathematicians tend to use words, and aerospace engineers tend to use acronyms (sometimes different acronyms for the same thing and some times the same acronyms for different things). But the airplanes are more likely to crash than are theorems.

7. MA = Martin's axiom. Stumbled over it once a while ago when i was floundering in the deep set theory ocean.

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