There is an interesting article in Quanta `Is Infinity Real?’ https://www.quantamagazine.org/20160630-infinity-puzzle-solution/ and this references an article in Discover `Infinity Is a Beautiful Concept – And It’s Ruining Physics’. I like both articles because they support my point about theories of statistical inference that anytime infinity is used we must be sure that we are in effect approximating something finite. This gets rid of a lot of paradoxical behavior. If you have a counterexample and it only holds when something is infinite, then you really don’t have a counterexample. A simple example of this occurs with the MLE. When the sample space and parameter space are finite, the MLE is consistent but of course there are many counterexamples where the MLE is not consistent and these depend intrinsically on infinity. So is the problem with the MLE or with the models? Certainly the mathematics is often much nicer when use is made of infinity, and that is fine by me, just don’t believe that models containing infinities represent the truth.

# Infinity does not exist

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I don’t think the issues are quite as simple as those articles make out. Concepts like ‘infinity’ are intrinsically linked to ideas like ‘continuity’ which are much harder to do away with mathematically or philosophically.

You might be interested in J.L. Bell’s book ‘The Continuous and the Infinitesimal in Mathematics and Philosophy’ which discussing these issues in depth. You can even download a copy from his website http://publish.uwo.ca/~jbell/The%20Continuous.pdf

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Thanks for the comment and the reference. Actually I have no problem with the concept of infinity for mathematics or philosophy and I can understand why it seems virtually necessary there. The problems arise when you go to apply it to real-world contexts and I think one has to be very careful and not get carried away. I can’t comment on physics, although I know many physicists have shared the opinion expressed in the article, e.g. E.T. Jaynes being one, but as far as statistical inference goes, all the data I have ever seen comes from some measurement process that is bounded and made to finite accuracy. As I pointed out in my comment, ignoring this can cause what seem like paradoxes but in reality are just artifacts of the mathematics as opposed to being real characteristics of the statistical problem. It is certainly fine to use statistical models that contain infinities but then remember that you are approximating something that is finite.

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Thanks for the reply.

I agree with almost all of what you say, but I prefer to think of ‘reality itself’ as ‘infinite’ – at least in a sense of ‘beyond our immediate grasp’ – while the results of measurements ‘of’ this reality as ‘finite’ (in a certain sense, anyway). So I like to think of an explicit measurement process from reality to numbers.

This leads to almost exactly the same positions on a number of topics as your view (I think), but perhaps some subtle differences.

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