- #1

- 91

- 14

here is the playlist in question

https://www.youtube.com/playlist?list=PL01A21B9E302D50C1

TIA

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- Thread starter hideelo
- Start date

- #1

- 91

- 14

here is the playlist in question

https://www.youtube.com/playlist?list=PL01A21B9E302D50C1

TIA

- #2

- 17

- 0

- #3

- 17

- 0

The real numbers are a totally ordered field with the least upper bound property (every non-empty set that is bounded above has a least upper bound). The Cauchy sequence or Dedekind cut constructions are existence proofs, not definitions. There is also a uniqueness proof.

Here's a brief informal overview of the Cauchy construction:

A sequence of rationals is a function ##s:\mathbb{N}\rightarrow\mathbb{Q}##. The ##n##th term, ##s_n=s(n)##. So we don't write these functions the same way as usual.

A Cauchy sequence is a sequence ##s## such that for any number ##\epsilon##, we can get ##\vert s_n-s_m\vert <\epsilon## for all the ##n,m\geq N## for some ##N##. In plain language, if we go far enough down the sequence, the rest of the points don't get further apart than ##\epsilon##.

Now, we take all the Cauchy sequences and we identify them as follows. Two Cauchy sequences ##t_n,s_m## are equivalent if given some ##\epsilon>0##, there is always some ##N## such that ##\vert t_n-s_m\vert<\epsilon## for all ##n,m\geq N##. In other words, if we far enough down both sequences, then all the remaining points are within ##\epsilon## of each other.

Since these definitions let ##\epsilon## be any positive rational number, it is helpful psychologically to think of ##\epsilon## being small.

These equivalence classes are going to be our real numbers. We define the operations of addition, multiplication, subtraction, and division of two equivalence classes by taking a sample Cauchy sequence from each, and then doing the respective rational number operation term wise.

It turns out by the triangle inequality that the resulting sequence of rationals is a Cauchy sequence, so it belongs to some equivalence class. It also turns out that no matter which sample Cauchy sequences you choose, you will end up in the same equivalence class at the end. The resulting structure satisfies the field axioms and takes just a little more work to verify that it satisfies the order properties we want.

Note that there are technically no limits or references to infinity used above. Also, we know that there are in fact Cauchy sequences of rationals (constant sequences for instance). An example of a non-constant Cauchy sequence of rationals is ##\frac{1}{n}##.

So far we would have the reals but no proof that there are non-rational reals.

For a cheap proof that there are Cauchy sequences of rationals with no rational limit (hence non-rational reals), the least upper bound of rationals that square to less than ##2## is a real number (by the axioms which are justified by the Cauchy construction), and it squares to ##2## (otherwise it would be too big or too small). There is a classic proof that no rationals square to ##2##.

Note that Wildberger avoids using technical language, as such he is prone to equivocating between what his words mean in common english versus their technical sense.

I just reread your post. Sorry that this is a bit tangential. :-(

Here's a brief informal overview of the Cauchy construction:

A sequence of rationals is a function ##s:\mathbb{N}\rightarrow\mathbb{Q}##. The ##n##th term, ##s_n=s(n)##. So we don't write these functions the same way as usual.

A Cauchy sequence is a sequence ##s## such that for any number ##\epsilon##, we can get ##\vert s_n-s_m\vert <\epsilon## for all the ##n,m\geq N## for some ##N##. In plain language, if we go far enough down the sequence, the rest of the points don't get further apart than ##\epsilon##.

Now, we take all the Cauchy sequences and we identify them as follows. Two Cauchy sequences ##t_n,s_m## are equivalent if given some ##\epsilon>0##, there is always some ##N## such that ##\vert t_n-s_m\vert<\epsilon## for all ##n,m\geq N##. In other words, if we far enough down both sequences, then all the remaining points are within ##\epsilon## of each other.

Since these definitions let ##\epsilon## be any positive rational number, it is helpful psychologically to think of ##\epsilon## being small.

These equivalence classes are going to be our real numbers. We define the operations of addition, multiplication, subtraction, and division of two equivalence classes by taking a sample Cauchy sequence from each, and then doing the respective rational number operation term wise.

It turns out by the triangle inequality that the resulting sequence of rationals is a Cauchy sequence, so it belongs to some equivalence class. It also turns out that no matter which sample Cauchy sequences you choose, you will end up in the same equivalence class at the end. The resulting structure satisfies the field axioms and takes just a little more work to verify that it satisfies the order properties we want.

Note that there are technically no limits or references to infinity used above. Also, we know that there are in fact Cauchy sequences of rationals (constant sequences for instance). An example of a non-constant Cauchy sequence of rationals is ##\frac{1}{n}##.

So far we would have the reals but no proof that there are non-rational reals.

For a cheap proof that there are Cauchy sequences of rationals with no rational limit (hence non-rational reals), the least upper bound of rationals that square to less than ##2## is a real number (by the axioms which are justified by the Cauchy construction), and it squares to ##2## (otherwise it would be too big or too small). There is a classic proof that no rationals square to ##2##.

Note that Wildberger avoids using technical language, as such he is prone to equivocating between what his words mean in common english versus their technical sense.

I just reread your post. Sorry that this is a bit tangential. :-(

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- #4

- 1,772

- 126

Zeilburger is a very good combinatorialist, though. He's not what I would call a crank. He just has some weird views is all.

- #5

- 91

- 14

- #6

- 1,772

- 126

- #7

- 91

- 14

Thanks again though

- #8

- 1,772

- 126

Thanks, in a perfect world I would go over every theorem on my own and see if I was convinced of it, and in reality I do try and do that to whatever extent possible. However when it comes to the assumptions that underlay set theory I wont be spending so much time on that, since I am really more interested in physics than in math, however I do not know where these unorthodox assumptions might have an effect. i.e. if one does not accept the axiom of choice are there far reaching consequences? I dont know.

I'm not sure they are that far-reaching because there's a whole constructivist school of thought that tries not to use the axiom of choice and gets pretty far with it. He does point out that a lot of mathematicians don't think about these foundational issues and don't understand them. It's interesting to think about, but when it comes down to it, even most mathematicians simply don't have time to wrap their heads around all the difficulties. That's the reason for specialization. Some people specialize in logic and foundations and we leave it to them to take care of and have all the debates. It is a bit irksome that mathematics has become so complicated that we don't have time to stop and think about it as much, though. That's one of the reasons I quit after my PhD. I felt like my intellect had no breathing room.

Similarly, if one does not accept the notion of infinity are there far reaching consequences mathematically?

In some ways, yes, but mainly that is just going to mean that mathematicians like him or Zeilberger are going to be limited to doing a certain kind of mathematics. As long as their arguments are correct, I don't see the problem. If you are worried about their approach, all you have to do is just take it as a supplement. Read something more standard to go along with it.

Im sure there are, but again I dont know. As a rule, I am more than happy to take math with plausibility arguments rather than building everything up from set axiomatic first principles. I dont know that to be necessary.

That seems to be conflating two different issues. Most people don't build it all from set theory. They start at a certain point and go from there. They have a certain starting point, but they may be very rigorous in terms of making complete, step by step arguments. Doesn't mean they understand the ultimate set-theoretic foundations. I think intuition is more important than logical proof, myself because intuition is what you are going to be able to take away and remember. Precise logic is very forgettable. If you have an intuitive idea of how it works, that's not really taking it on faith, as I see it. You should have some appreciation for how intuition can fail, though, and always be careful.

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