If something moves at an infinite speed how long does it take to traverse infinity?

If the answer is infinite time, then is there something greater than infinity such that something can traverse infinity in 0 seconds or less than 1 second?

• How infinite a speed and how infinite a distance to traverse? "Infinity" has infinitely many gradations, it can be anything from 0 to any one of them if we assume that non-standard models with infinite numbers describe some kind of physics. – Conifold May 2 at 20:08
• In relativistic physics, there isn't a meaningful notion of infinite speed. Also, in general, the time it takes to traverse a distance is given by distance/speed, which in this case would be ∞/∞, which is an indeterminate form. – Sandejo May 2 at 20:59
• Or it might be ω/ω in surreal numbers, which would be 1 iirc. – Kristian Berry May 2 at 21:04
• In physics the final observables must be finite and measurable, any infinite observable such as time and distance here is considered an anomaly and your theory needs to be renormalized unless you make them as hidden variables in your theory, then you can use transfinite math to freely imagine and construct using ω or cardinals... – Double Knot May 2 at 21:35
• Basically you are asking "What is infinity divided by infinity?", which has an answer here: math.stackexchange.com/questions/181304/… – kutschkem May 3 at 8:54

This is not a very well-defined question. Essentially, you are asking to compare two infinities, that is, to find their ratio: inf/inf.

In general, the answer -- well known to mathematicians -- is to examine the processes that caused you arrive at infinity -- to examine the limit:

lim (x -> inf, y -> inf) x/y = ?

Without further specifying what kind of infinities these are, that is, how you got to them, inf/inf is an undefinable quantity.

p.s. can anybody get math rendering to work?

Suppose you were at the start of a road that went on for an infinite number of miles. If you are travelling infinitely many miles per hour (but only per hour!), it will take you an hour to reach the end of the road. Generally, however long it takes you to travel an infinite number of miles, is how long it will take you to get to the end of the road.

To answer your question about something "greater than infinity," let us compare two roads, one ℵ0-miles-long and another ℵ1-miles-long. Since ω×ω and ωω (for example) are less than ω1, infinitely so, if you are travelling at ℵ0 miles per hour, even after ω hours you will not reach the end of the second road. Now, if your speed multiplied your distance per hour, that is if you travelled at some rate that mapped into ℵ00, then after ω hours, you would at least reach the end of the second road, if not an even longer one.

I think.

(Actually, I'm not sure the above is worded correctly, or if it even can be worded correctly. The best I can say is that something like the above would have to be true, if your question is to have an interesting answer.)

• If you're going infinitely many miles per hour, then you are also going infinitely many miles per microsecond. To see this, consider the contrapositive. If you travel a finite number of miles per microsecond then you also travel a finite number of miles per hour. Your analysis doesn't work. The rest of your post is confusing since you seem to be mixing ordinals and cardinals and the argument's unclear to me. Are you thinking of the long line? en.wikipedia.org/wiki/Long_line_(topology) – user4894 May 5 at 0:02
• In fact if you travel infinitely fast, you can travel any finite distance in zero time; hence any countably infinite distance in zero time. To go an uncountable distance you're adding up uncountably many zeros, which might be zero or it might not. For example the integral of dx over the unit interval is 1, so the sum of uncountably many zero-width vertical slices is nonzero. All in all I don't think your argument works. – user4894 May 5 at 0:10
• I knew there was something wrong with what I said. Argh! – Kristian Berry May 5 at 0:12
• Until we can solve CH (continuum hypothesis) under some enriched math foundation from ZFC like system or physical spacetime turns out to be discrete like PA or ACA like systems, we know very little how to apply transfinite numbers to physics even as hidden variables I guess... – Double Knot May 5 at 1:35
• If one takes speed to just be change in position over change in time (in some specified inertial coordinate system) between different points on a worldline, then infinite speed would just be a line of constant t-coordinate, so the coordinate time between any two points on this line should be zero even if you extend the spatial axis to infinite hyperreal numbers or something. Maybe the time between points an infinite distance apart could be finite if the slope in an x-t coordinate system was infinitesimal rather than zero, since the hyperreal system does allow for nonzero infinitesimals. – Hypnosifl May 5 at 3:35