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March 6th, 2018, 05:26 AM   #21
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Unfortunately you've stopped listening to what I say. You are arguing against straw men that run counter to points I have made explicitly. I'll give it one last go.
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Originally Posted by Karma Peny View Post
As I said, to say that these things exist because they can be imagined is equivalent to saying unreal things must exist because someone says (s)he can imagine them.
The mathematical universe is imagined and everything in it. You yourself mentioned that neither circles nor straight lines actually exist, yet you allow them in your world. In mathematics things exist that have a definition.

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Originally Posted by Karma Peny View Post
The human brain is a finite structure and so it can only process finite quantities.
You can't imagine what 100 billion looks like either, but that doesn't stop 100 billion things existing. If you have any concept at all of 100 billion grains of sand, for example, it will be in terms of a few smaller piles of sand. And that will tell you nothing about 100 billion coins.

Everthing we know of the infinite is the result of logical deduction, just like our knowledge of the cosmos and sub-atomic particles.

In point of fact, nothing here requires the infinite. We only need the natural numbers. If you accept that they are unbounded (whether or not you accept the infinite), the rest is pure logic.
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Originally Posted by Karma Peny View Post
Rather than trust our imagination or our intuition, we could examine what we can represent with physical materials.
This is a dumb argument. The US government has trillions of dollars of debt. Nobody has ever had trillions of things to manipulate in order to very anything.

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Originally Posted by Karma Peny View Post
There are many other series that do not have similar geometric derivations and so this in no way refutes my main argument.
Except that, if you admit $\pi$ and $\sqrt2$, you have opened Pandora's box. Numbers exist that do not have a finite decimal representation. From there, the least disruptive move is to accept all non-terminating decimals. This is especially true once you take the definition of the real numbers as the limits of (non-terminating) rational sequences. I use "non-terminating" to align with Wildberger's characterisation of the natural numbers. Thus we avoid any worries about the infinite.

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How can you find an infinite limit without doing infinite things?
From the definition. You pick an $\epsilon$, I give you an $N$ and it doesn't matter which $n \gt N$ you try (or how many), I guarantee that the $n$th element of the sequence is within $\epsilon$ of the limit.

There is no need for infinite actions because the logical steps allow us to aggregate over huge domains. It's no different to saying that $x^2 \ge 0$ for all $x$.

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Originally Posted by Karma Peny View Post
I can only presume that you find it a completely satisfactory state of affairs that the limit of a pi sequence can only be defined in terms of another pi sequence.
This just ignores what I've said.

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Originally Posted by Karma Peny View Post
I show that if they are one and the same thing then they cannot equal 1.
No you don't. In fact, your argument denies talking about real numbers by denying that they are limits rather than "infinite sums".

Here's a question: how can you claim any knowledge of "infinite sums" if you deny that anyone can know anything about the "infinite"?

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Originally Posted by Karma Peny View Post
The best we can do is line up terms in a way that we hope will all cancel out, all the way to infinity, as is supposedly done in the algebraic proof for 0.999... = 1.
No. I reject that characterisation of the proof. It doesn't describe the mathematics of what it happening.

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And yet you disagree that limit is the sum of its terms.
No, I don't. A limit is a limit. It's a number. A sequence is not a number.

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You are grasping at straws to avoid admitting that a limit must be the addition of its terms by necessity.
Not at all. A limit has a definition that doesn't refer to summation and doesn't even refer to the end point. A limit expressly says nothing about an "infinite sum".
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March 6th, 2018, 05:38 AM   #22
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Originally Posted by Micrm@ss View Post
Why can't I define 2.000000... as the real number?
Because it's just a representation of the number. The number is the abstract thing that sits behind the representations $2,\frac63,2.000\ldots$ or the calculation of the limit of a particular sequence.

The representation 2.000 doesn't mean anything without having a mechanism to assign a value to it. And simply saying "we add up all the terms of the implied series" doesn't work because we can't add infinitely many terms. The best we can do is to calculate the limit of the partial sums. Therefore we arrive at the point where the limit is the number.

Last edited by v8archie; March 6th, 2018 at 05:42 AM.
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March 6th, 2018, 08:15 AM   #23
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Quote:
Originally Posted by v8archie View Post
But a better (and equivalent) definition is that every real number is the limit of at least one sequence of rationals.
I'm a little confused by this: it seems like you're defining real numbers in terms of something that itself needs to be defined. In particular, how are you defining limits of cauchy rational sequences that don't have rational limits?

Last edited by cjem; March 6th, 2018 at 08:25 AM.
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March 6th, 2018, 08:40 AM   #24
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I don't understand how you can use the term Cauchy sequences and not know how their limits are defined?
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March 6th, 2018, 09:09 AM   #25
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Quote:
Originally Posted by v8archie View Post
I don't understand how you can use the term Cauchy sequences and not know how their limits are defined?
The definition I've always used is that the limit of a cauchy sequence $(a_n)$ of rationals is the real number $a$ such that for all $\epsilon > 0$ blah blah blah. As this definition makes reference to the reals, it certainly can't be used to define them!

We can certainly define what it means for rational sequences to have a rational limit without mentioning the reals. But then the issue is that most cauchy sequences of rationals (and all of the ones we're interested in here) don't actually have limits in this sense.

Do you have a definition for the limit of a rational sequence that doesn't reference the reals and that leads to every cauchy sequence having a limit? I suspect to come up with one and make sure it's meaningful, you'd have to essentially do the hard work of any other construction of the reals. But I'd be very happy if you could show me wrong!

Last edited by cjem; March 6th, 2018 at 09:28 AM.
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March 6th, 2018, 10:11 AM   #26
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Quote:
Originally Posted by cjem View Post
The definition I've always used is that the limit of a cauchy sequence $(a_n)$ of rationals is the real number $a$ such that for all $\epsilon > 0$ blah blah blah. As this definition makes reference to the reals, it certainly can't be used to define them!

Do you have a definition for the limit of a cauchy sequence of rationals that doesn't make reference to the reals? We can certainly define what it means for rational sequences to converge in $\mathbb{Q}$ without mentioning $\mathbb{R}$, but then none of the sequences we're interested in actually have limits under this definition!
Say we have the rationals but not the reals, and we don't like Dedekind cuts. How can we construct the reals another way?

Let $(s_n)$ and $(t_n)$ be Cauchy sequences of rationals.

Suppose that for all $\epsilon$ there exists $N$ such that for all $m, n > N$, we have $|s_m - t_n| < \epsilon$.

This is an equivalence relation on rational Cauchy sequences, and the set of equivalence classes are ... voilĂ ! The reals. All this needs proof of course, just epsilon chasing.

https://en.wikipedia.org/wiki/Constr...uchy_sequences

Last edited by Maschke; March 6th, 2018 at 10:20 AM.
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March 6th, 2018, 10:16 AM   #27
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Quote:
Originally Posted by Maschke View Post
Say we have the rationals but not the reals, and we don't like Dedekind cuts.

If we have two Cauchy sequences of rationals, their term-by-term difference may itself be Cauchy; that is, for all $\epsilon$ there's a large $N$ such that all the term-by-term differences are less than $\epsilon$ past $N$.

This is an equivalence relation on Cauchy sequences, and the set of equivalence classes are ... voilĂ ! The reals.

https://en.wikipedia.org/wiki/Constr...uchy_sequences
That's essentially the construction of the reals I like the most (in fact, I prefer the very similar construction of taking the reals to be the ring of cauchy rational sequences quotiented out by the maximal ideal of null sequences - it gives a few nice properties for free, such as that the reals are a field).

If you notice, however, v8archie proposed an alternative definition (see what I quoted in post #23) to this - I'm simply trying to see if it holds any water.

Last edited by cjem; March 6th, 2018 at 10:28 AM.
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March 6th, 2018, 10:22 AM   #28
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And all sequences in the equivalence class have the same limit: a real number. Thus every limit of a Cauchy sequence of rationals is a real number.
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March 6th, 2018, 10:25 AM   #29
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Quote:
Originally Posted by cjem View Post
If you notice, however, v8archie proposed an alternative definition and I am simply trying to see if it holds any water.
Oh I see. v8archie said:

Quote:
Originally Posted by v8archie
But a better (and equivalent) definition is that every real number is the limit of at least one sequence of rationals.
Of course that's incorrect. You can't assume the existence of the reals to define the reals. Prior to the construction of the reals, Cauchy sequences of rationals often don't have limits. [I know you know this, my comment is for v8archie].

In other words if we have the rational Cauchy sequence $1, 1.4, 1.41, 1.414, \dots$, we would LIKE to say but we CAN NOT say "Well $\sqrt{2}$ is just that limit, can't you see??" We can't say that because there is a hole in the rationals right where we imagine there should be a real. [Again, for v8, not you].

It is a theorem that every real is the limit of a sequence of rationals, once you've defined the reals purely in terms of the rationals.

Last edited by Maschke; March 6th, 2018 at 11:00 AM.
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March 6th, 2018, 10:53 AM   #30
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Quote:
Originally Posted by v8archie View Post
And all sequences in the equivalence class have the same limit: a real number. Thus every limit of a Cauchy sequence of rationals is a real number.
The issue is that the usual definition of a limit makes use of real numbers (at least implicitly). So defining real numbers in terms of these limits is completely circular.
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