{"items": [{"author": "Jeff Kaufman", "source_link": "https://plus.google.com/103013777355236494008", "anchor": "gp-1423321422041", "service": "gp", "text": "@Ben\n\u00a0\n@Shawn\n\u00a0", "timestamp": 1423321422}, {"author": "Daniel", "source_link": "https://plus.google.com/104241554778763268733", "anchor": "gp-1463362628644", "service": "gp", "text": "As others were indicating, axiom #1 isn\u2019t always bidirectional, and this is one of those cases.\n
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\nIn this explanation, I use \u03c9 as the smallest infinite ordinal number (see \nhttps://en.wikipedia.org/wiki/Ordinal_number#Ordinals_extend_the_natural_numbers\n). For example, in (9)4, the first digit past the decimal is 9, the second is nine, and for any natural number n, digit n after the decimal place is 9. However, digit \u03c9 after the decimal is 4, digit \u03c9 + 1 is 0 (by axiom 4), and all subsequent digits after the decimal are also 0.\n
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\nAfter the first so in your proof, you say\n
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\n(x)x = (x) by #1.\n
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\nThis is false, and does not follow from before. Consider the example of x = 9. The part before the \u201cso\u201d has been dealing with ((9)9) = ((9)). With just the first repetition, this means that for all natural numbers n, the nth digit after the decimal is 9. Adding the second, we get that for all n and m, digit \u03c9\u00b7n+m is nine. However, digits \u03c9\u00b2 and beyond are all zero (which is why ((9)) < ((9))1). Next, you try to use property #1 to get that (9)9 = (9). But this doesn\u2019t work: in the former, digit \u03c9 is 9, while in the latter it\u2019s 0.\n
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\nI believe this is related to the fact that \u03c9 + 1 > \u03c9, but 1 + \u03c9 = \u03c9. Infinity is counterintuitive; tread with caution.", "timestamp": 1463362628}, {"author": "Daniel", "source_link": "https://plus.google.com/104241554778763268733", "anchor": "gp-1463362941696", "service": "gp", "text": "On an unrelated topic, expand the ASCII-ized notation to include other forms of brackets. Then you could write [(9)9] instead of ((9)9). This helps keep track of which overline you\u2019re representing.\n
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\nIn all examples so far, this doesn\u2019t help, because the overlines have been nested. But do you have any idea what 0.1[2(3]4)5 could mean?", "timestamp": 1463362941}]}