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I was talking to a group of more or less advanced graduates in one forum. One of them claimed that the equality of two real numbers is undecidable.

I agreed since, if we represent real numbers just as infinite strings without further context, the only option is to check each digit. However, I added, if we can compare their algorithms behind those infinite strings, maybe the equality could be decidable.

To my surprise, he denied my statement and said that even if real numbers were represented as algorithms, the equality still would be undecidable. He refused to elaborate and at least give me some literature to read about it.

If his claim is indeed true, could you, please, provide me with some papers or books, where i could read these results on my own?

Personally, I couldn't find any thorough proof of the undecidability of equality, however every book states it too.

Maulsharif Aidos
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    See this duplicate. And Richardson's Theorem – lulu Mar 23 '22 at 12:01
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    We cannot decide in general whether a function computed by an algorithm has any given nontrivial property (for example increasing , constant , outputs are all prime numbers). In particular, we cannot decide whether two algorithms compute the same function and the digits of a real number can be considered to be such a function. – Peter Mar 23 '22 at 12:17
  • It's an intetesting fact that while equality of real numbers is undecidable, inequality is semi-decidable. This turns out to be profoundly related to the fact that $\Bbb R$, with its usual topology, is a Hausdorff space but not a discrete one. – MJD Mar 23 '22 at 18:25

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