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Wouldn't a set of numbers that is ordered like

$$0.1,0.2,\ldots,0.9,0.11,0.12,0.13,\ldots,0.99,0.101,0.102,\ldots$$

(skipping values that repeat such as $0.10$, $0.100$, etc.)

necessarily include all values between $0.1$ and $1$ as a countable infinite? Since you could match each value after the decimal to a value on the countably infinite integer line.

Of course, I don't think that this is some "new discovery" or anything. I'm just trying to find a proof or example that demonstrates that this is still uncountably infinite.

Blue
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twalker
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2 Answers2

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The decimals that terminate (i.e. finitely many nonzero digits) are countable, and this is what you show.

Note, however, that you miss lots of numbers, such as all the irrationals and even some rational like $1/3$ (more generally you miss every rational whose denominator is a multiple of some prime $p \neq 2,5$ when written in reduced form)

Brian Moehring
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    Thank you for responding! I am writing this as a response to everyone who mentioned irrationals, $\frac{1}{3}$, etc..

    I am not disagreeing with you, I am merely confused as to how this method does not include values such as $\frac{1}{3}$, since it would be written out as 0.33333333..., wouldn't this just match to the integer 33333...? I think my confusion stems from my idea of a countable infinity, since in my head it makes sense that the set of all integers includes any repeating value like 11111...,33333..., etc. Could you please explain how the set of all integers does not contain this?

     – twalker Aug 26 '20 at 23:45
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    33333... is not a number. – markvs Aug 26 '20 at 23:52
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    what do you mean by it's not a number?Maybe I got it, it is not the antecedent of any number – Tortar Aug 26 '20 at 23:56
  • Why is it not? If the set of all integers contains integers with lengths that go on infinitely, wouldn't this include 333333...? – twalker Aug 26 '20 at 23:57
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    Who told you that integers can go on infinitely? That is not allowed. Real numbers can have infinitely many non-zero digits, but integers can’t. – Thomas Andrews Aug 26 '20 at 23:59
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    @twalker The lengths of the integers may be arbitrarily large, but each is still finite length. No integer has infinite length. – Brian Moehring Aug 27 '20 at 00:00
  • Ah, I think that's where I was confused. I had incorrectly assumed that integers could go on infinitely since the set of integers is infinite. My logic was that there is always going to be an integer that is one digit longer than the previous, so this would go on infinitely. Thank you for the correction. – twalker Aug 27 '20 at 00:01
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    I think we both can learn from here – Tortar Aug 27 '20 at 00:16
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This set does not include $\frac13$.

markvs
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