Infinity Today

Mathematical practice obtains information about what it would be like if there were infinitely many things. It is not at all evident a priori that we can do that. But mathematicians have shown us that we can. Our steps are tentative, but as confidence is acquired we move forward. Our theorems are proved in many different ways, and the results are always the same. Our formalisms are robust and yield information beyond the original intent. To doubt the significance of the concrete evidence for the objectivity of mathematical knowledge is like anti-evolutionists doubting the evidence of paleontology by suggesting that those fossils were part of creation. As was discussed above, Godel’s work has left us with a transfinite sequence of formal systems involving larger and larger sets. Models of these systems can be obtained from initial segments of the famous hierarchy obtained by iterating transfinitely the power set operation P:

Thus, Vm2 is a model of the original Zermelo axioms. To obtain a model of the more comprehensive Zermelo-Fraenkel (ZF) axioms, no ordinal whose existence is provable in ZF will do.[1] To continue the transfinite sequence of formal systems, it is necessary to enter the realm of large cardinals in which there has been intensive research. Workers in this realm are pioneers on dangerous ground: although we know that no proof of the consistency with ZF of the existence of these enormous sets is possible, it is always conceivable that a proof in ZF of the inconsistency of one of them will emerge thereby destroying a huge body of work. But the empirical evidence is encouraging. Although the defining characteristics of the various large cardinal types that have been studied seeming quite disparate, they line themselves up neatly in order of increasing consistency strength. Moreover, they have shown themselves to be the correct tool for resolving open questions in descriptive set theory.

So far Godel incompleteness has had only a negligible effect on mathematical practice. Cantor’s continuum hypothesis remains a challenge: although the Godel- Cohen results prove its undecidability from ZF, if the iterative hierarchy is taken seriously, it does have a truth value whether we can ever find it or not. In the realm of arithmetic many important unsolved problems, including the Riemann Hypothesis and the Goldbach Conjecture, are equivalent to П0 sentences. However, so far no undecidable П0 sentences have been found that are provably equivalent to questions previously posed (as has been done for uncomputability). However, Harvey Friedman has produced a remarkable collection of П0 and П20 arithmetic sentences with clear combinatorial content that can only be resolved in the context of large cardinals.

  • [1] Because otherwise the consistency of ZF would be provable in ZF contradicting Godel’s secondincompleteness theorem. For that matter the set Vm2 cannot be proved to exist from the Zermeloaxioms alone; in ZF its existence follows using Replacement.
 
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