Decimal Inconsistency

Ben Orlin and I were playing around with a weird kind of infinite repeating decimal. He wrote up a blog post, there was some discussion, but now I think there's a contradiction in what I thought were reasonable axioms.

First a summary. This is all based around the idea that you can have "0.9̅3" or "0.9̅4," and in fact "0.9̅3 < 0.9̅4". The first one is 0.999...3 while the latter is 0.999...4. Or "first you have nines forever, and then either a three or a four". Since 3 < 4, we should have 0.9̅3 < 0.9̅4. If this is confusing Ben's post goes into more detail.

(I'll note here that this isn't normal math. You can't add these, subtract them, multiply, etc. Normally 0.9̅ is exactly 1 and 0.9̅3 is meaningless. We're playing with some things that are kind of like numbers, but not entirely.)

Here are some properties it seems like these numbers should have, where x and y are infinite decimals and R is any of >, =, or <. To simplify writing in text we're writing 0.x̅y as (x)y.

1. x R y → (x) R (y)

2. x R y → xz R yz

3. x R y → zx R zy

4. x = x0

5. x(x) = (x)

6. (xy) = x(yx)

Here's the contradiction, which my coworker Shawn figured out:

  ((x)x) = (x)(x(x))   by #6
         = (x)((x))    by #5
         = ((x))       by #5

so

    (x)x = (x)         by #1
         = (x)0        by #4

so

       x = 0           by #3

which is a contradiction.

This seems right to me, but all of the axioms also seem reasonable. I'm not sure what you would drop to make this more reasonable.

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