1 Answer
...whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement )
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one $\begingroup$ This makes sense, thank you for your answer. What about rational numbers? I guess you could write them as pairs, but was it actually done? $\endgroup$ – user507115 Commented Jun 14 at 20:01 -
one $\begingroup$ @richardIII Yes, rational numbers - even algebraic numbers - are no more complicated than naturals or integers in this sense. I don't know a particularly good source on this, but the basics are surely folklore (e.g. that EFA proves, appropriately formalized, the statement "the rationals from a dense linear order without endpoints" and so on). $\endgroup$ Commented Jun 14 at 20:06 -
one $\begingroup$ Maybe worth noting that many theorems we recognize as theorems about real numbers can be phrased purely in terms of rationals. For example, "$\sqrt{2}$ is irrational" can be phrased as the equivalent statement "There is no rational number $x$ such that $x^2=2$" And more subtle ones can also be phrased this way. One can even make a statement equivalent to "Pi is transcendental" in EFA, albeit in a pretty convoluted way. $\endgroup$ – JoshuaZ Commented Jun 15 at 1:38 -
four $\begingroup$ Reals, sequences of reals, or continuous real functions can be defined in $\mathrm{RCA}^*_0$, which is a conservative extension of EA (that Friedman calls EFA). $\endgroup$ Commented Jun 15 at 6:05 -
one $\begingroup$ @EmilJeřábek CODES for continuous real functions can be defined there. $\endgroup$ Commented Jun 15 at 9:23