skipjack

There's no justification for that wording, as you haven't yet shown that there is such a thing as the list of real numbers. You stated "There is no "list at infinity".

As there is no "list at infinity", that doesn't imply that the reals are countable.

Suppose instead that you make a list of n identical sequences, all of length 10, each sequence being 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. You can choose a single appropriate digit from these sequences so that your chosen digits are 7, 0, 7, 1, 0, 6, 7, . . . continuing until n digits have been listed. By continuing in this fashion (with increasing values of n), your choices can form the first n places of the decimal expansion of the square root of 1/2, or any other real in [0,1). You need a countable number of steps to do this, so by your logic you could claim that the reals in [0,1) are countable. That would lack proper justification, as being able to take an arbitrarily large, but finite, number of steps towards some goal doesn't imply that the goal is itself countable (in any sense). Even if you took an infinite sequence of steps towards the goal, that still wouldn't prove that the goal is countable.

zylo

That's what I said back on pg 4, now buried, by misrepresentations, delierately ignoring what I said, or irrelevancies based on not understanding what I said.

There is no list of real numbers at infinity (a number), but you can talk about an infinite (unending) list. The list of n-place rational numbers gets closer and closer to the list of real numbers and remains unique as n gets larger and larger. That's the fundamental concept of real analysis.
If a property is true for all (any) n, in this case uniqueness and countability, it is true for countably infinite n.

As for the second part of previous post, I find the example irrelevant and the conclusion unintelligible.

Benit13

No... we are not ignoring what you said, it's just that your argument is flawed and the only responses you are giving to the objections are just reassertions of other flawed things you have said earlier.

Well, maybe that's where you need to go and study!

Also... your scheme to list all of the real numbers doesn't contain all of the real numbers (e.g. 1/3 is not in your list). It can only contain that number if n is allowed to be infinity (not just approach it), but you already objected to that.

skipjack

You referred to the specific unending list of all the real numbers in [0,1), and did so before showing that such a list exists.

That doesn't imply the countability of the irrationals, as all your countable lists contain only decimals corresponding to rational values. You can use a selection process to find listed rationals that approach any irrational real number in [0,1), but that doesn't show that those reals are countable, because each finite selection of rationals gets used for such an approach to infinitely many reals (that you haven't shown are countable).

v8archie

You are not doing real analysis here, you are trying to do set theory. Real analysis doesn't care how many reals there are. Neither are lists any part of real analysis except when interpreted as sequences.

Moreover, when $n$ gets larger and larger, there is no suggestion in real analysis that this has anything to do with "infinity". It just means that $n$ can become arbitrarily large.

Related to your discussion, this just means that your $n$-place decimals can become arbitrarily long, but by virtue of having a length they remain finite.

This is not generally true. The $n$-place decimals are all of finite length and are all rational numbers. Neither of these properties hold for infinite decimals. Your claims (uniqueness and countability) are no more than unproven (and untrue) assertions. Moreover, there is no mechanism by which you can increase $n$ sufficiently that it becomes, or even "approaches" an infinite value.

The concept of limit in real analysis applies to the values of the elements of a sequence. In particular, the sequence converges to the value $L$ only if for any (arbitrarily small) value $\epsilon$ we can always find an $N$ such that the value of every element of the sequence after the $N$th is within $\epsilon$ of $L$. At no point does any concept of "infinity" get involved in this definition. You cannot use this to to tell you anything about set theoretic infinities.

zylo

My posts, buried in this thread, are summarized and generalized, simply and clearly, in:

Decimal Representation and Expansion

v8archie

So the sum total of your posts here is that you believe, on no evidence whatsoever, that whatever is true for finite cases must also be true for infinite cases.

Moreover, you prefer to ignore the evidence presented to the contrary, instead repeating your bogus claim as if nobody had commented at all.

It doesn't matter how many times you repeat nonsense, it remains nonsense.

skipjack

The thread you linked to doesn't give any mathematical argument at all; it does little more than define a term that you haven't used in any of your posts here.

In particular, you don't explain what you mean by "limit" here, and you cannot claim to have proved anything here if you don't explain what you mean by that.