#### Mathematical Infinity

We might well believe that our physical universe is finite; but mathematics appears to posit infinite entities, such as the collection of the natural numbers. This raises a fundamental question in the philosophy of mathematics:

*To what extent and in which sense do mathematical infinities really exist?*This question can be broken up in the following sub-questions:

- Do potentially infinite collections exist?
- Do actually infinite collections exist?
- Do the actually infinite collections themselves form a potentially infinite collection of infinities of different sizes?
- Is there a maximally large actual infinity?

I will argue that it is rational to answer ‘yes’ to each of these questions.

Advertisements
(function(){var c=function(){var a=document.getElementById("crt-528024105");window.Criteo?(a.parentNode.style.setProperty("display","inline-block","important"),a.style.setProperty("display","block","important"),window.Criteo.DisplayAcceptableAdIfAdblocked({zoneid:388248,containerid:"crt-528024105",collapseContainerIfNotAdblocked:!0,callifnotadblocked:function(){a.style.setProperty("display","none","important");a.style.setProperty("visbility","hidden","important")}})):(a.style.setProperty("display","none","important"),a.style.setProperty("visibility","hidden","important"))};if(window.Criteo)c();else{if(!__ATA.criteo.script){var b=document.createElement("script");b.src="//static.criteo.net/js/ld/publishertag.js";b.onload=function(){for(var a=0;a<__ATA.criteo.cmd.length;a++){var b=__ATA.criteo.cmd[a];"function"===typeof b&&b()}};(document.head||document.getElementsByTagName("head")[0]).appendChild(b);__ATA.criteo.script=b}__ATA.criteo.cmd.push(c)}})();
(function(){var c=function(){var a=document.getElementById("crt-57030387");window.Criteo?(a.parentNode.style.setProperty("display","inline-block","important"),a.style.setProperty("display","block","important"),window.Criteo.DisplayAcceptableAdIfAdblocked({zoneid:837497,containerid:"crt-57030387",collapseContainerIfNotAdblocked:!0,callifnotadblocked:function(){a.style.setProperty("display","none","important");a.style.setProperty("visbility","hidden","important")}})):(a.style.setProperty("display","none","important"),a.style.setProperty("visibility","hidden","important"))};if(window.Criteo)c();else{if(!__ATA.criteo.script){var b=document.createElement("script");b.src="//static.criteo.net/js/ld/publishertag.js";b.onload=function(){for(var a=0;a<__ATA.criteo.cmd.length;a++){var b=__ATA.criteo.cmd[a];"function"===typeof b&&b()}};(document.head||document.getElementsByTagName("head")[0]).appendChild(b);__ATA.criteo.script=b}__ATA.criteo.cmd.push(c)}})();