In a very reasonable sense, the real numbers do not exist in the real world. Almost all real numbers are non-computable, so under apparently reasonable assumptions about what experiments you can conduct, there is no experiment you can do that will produce a measurement with the value of most real numbers.

From this, it’s fairly non-controvesial to say that only the computable reals exist; these are a tiny (measure-zero) subset of the reals.

If you go further and assume a fundamentally discrete universe (much more controvesial), then all you can really measure are integers.