The Largest Number With A Name: Fill & Download for Free

The largest finite number with a name is probably Rayo's number but, if you look at its definition, you will see there is little point in defining such numbers. It isn't even immediately obvious that it is well-defined, unlike the similar Berry paradox which is not well-defined. And do we get a different number if we allow two googol symbols rather only one googol? Personally, I don't know (but I think not).It is of course trivially small compared with almost all Natural numbers, none of which have a name, and infinitesimal compared to the smallest limit Ordinal, named [math]\omega[/math] (pronounced "omega", the last letter of the Greek alphabet) or the smallest transfinite Cardinal, named [math]\aleph_0[/math] (pronounced "aleph null" or "aleph naught", the first letter of the Hebrew alphabet with a subscript of zero).For every named Ordinal number [math]\lambda[/math] there is a named Cardinal number [math]\aleph_{\lambda}[/math]. Even [math]\aleph_1[/math] causes trouble (see the continuum hypothesis) for our usual collection of foundational axioms, ZFC. And you definitely don't want to go beyond [math]\aleph_{\omega}[/math] without a deep understanding of cofinality with which you may be able to travel to the land of truly large numbers where reside inaccessible Cardinals, admissable Ordinals, and so on.The largest named Cardinal number that makes any real sense to me is [math]\aleph_{\omega_1}[/math] where [math]\omega_1[/math] is the first uncountable Ordinal (and you should bear in mind that Ordinals like [math]\omega^{\omega}[/math] are perfectly countable).

I believe it is Graham's number. It is so big that you really need a special notation just to write it down. The ordinary decimal notation would have more digits than there are atoms in the observable universe.SeeGraham's number

Scott Aaronson's essay on this subject, "Who Can Name the Bigger Number?" is a true masterwork, giving substance and coherence to this seemingly jokey idle question and connecting it to some of the deepest ideas in computer science and discrete mathematics more generally.http://www.scottaaronson.com/writings/bignumbers.htmlIt begins:In an old joke, two noblemen vie to name the bigger number. The first, after ruminating for hours, triumphantly announces "Eighty-three!" The second, mightily impressed, replies "You win."A biggest number contest is clearly pointless when the contestants take turns. But what if the contestants write down their numbers simultaneously, neither aware of the other’s? To introduce a talk on "Big Numbers," I invite two audience volunteers to try exactly this. I tell them the rules:You have fifteen seconds. Using standard math notation, English words, or both, name a single whole number—not an infinity—on a blank index card. Be precise enough for any reasonable modern mathematician to determine exactly what number you’ve named, by consulting only your card and, if necessary, the published literature.So contestants can’t say "the number of sand grains in the Sahara," because sand drifts in and out of the Sahara regularly. Nor can they say "my opponent’s number plus one," or "the biggest number anyone’s ever thought of plus one"—again, these are ill-defined, given what our reasonable mathematician has available. Within the rules, the contestant who names the bigger number wins.