An interesting point but I&#x27;m not sure that it does. The ultra-finitist says that numbers above a certain limit don&#x27;t exist. In my view numbers don&#x27;t &quot;exist&quot; in the first place. The theory of numbers is a tool for making predictions. The statement that numbers above a certain limit don&#x27;t exist turns into the statement that <i>examples to test the theory</i> don&#x27;t exist above a certain limit, because you won&#x27;t be able to find that big of an example in our universe. IMO this is a much more sensible statement. An ultra-finitist may worry about an intermediate number in a calculation not existing, even though the start and end number may be small (for instance consider log_2(2^n) = n). This is silly in my view, since the theory of numbers is a prediction tool as a whole, and there is no need to physically realize intermediate numbers. As long as you&#x27;re using the rules of the theory you&#x27;re good (for the same reason you don&#x27;t need to be afraid of using infinite sets in the middle of a deduction). One might also worry that the theory of numbers may fail to make correct predictions for very large numbers, but this is really a different question about generalising from examples. It could already fail for very large examples that we <i>can</i> test in our universe. The possibility of it failing for numbers larger than we can realize in our universe is precisely what you <i>don&#x27;t</i> need to worry about.