Re: Does the number of nines increase?

After serious thinking Peter Fairbrother wrote :

On 04/07/2024 21:01, FromTheRafters wrote:

Chris M. Thomasson laid this down on his screen :

On 7/4/2024 9:58 AM, FromTheRafters wrote:

Peter Fairbrother formulated on Thursday :

On 25/06/2024 21:18, WM wrote:

Let the infinite sequence 0.999... be multiplied by 10. Does the number of nines grow?
Corollary-question: Does the number of nines grow when in 0.999 the decimal point is shifted by one or more position?
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Regards, WM

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A more interesting question; suppose a set containing an infinite number of 9's. Now copy that set and add a 2.
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Is the second set bigger than the first set?
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Peter Fairbrother

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Lrf, pneqvanyvgl bar naq pneqvanyvgl gjb.

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Different set with an infinite number of elements. Their sizes are the same for infinity = infinity... :^)

 Providing one infinite set and the other are both countable or both uncountable. If one set is size Aleph_zero and the other is 2^Aleph_zero then they are not the same size.

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I imagined the 9's as being the last digits of 0.99... so distinguishable and countably infinite.

There is nothing wrong with a sequence having duplicate members. Sets (ZFC) don't have duplicates though.

But the second set is also countable.

Yes, a plus one to either infinity doesn't change the size of the infinite set.

So the sets have the same cardinality, (?and the same size?), but the element 2 is in one set but not in the other, and no other element is in one set but not the other.
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Peter Fairbrother

A difference between the idea of 'same' and 'equal size'. 'Same' if each can be a subset/superset of the other (matching elements) and 'equal size' in terms of cardinality (pairing elements).