User blog comment:Vel!/Limit of Communication/@comment-30754445-20170623160435

Too really make this question interesting, the limit should be much less than a googol bits.

Because it really isn't too difficult to transmit the entire body of human mathematical knowledge with a much much smaller number of bits.

Here's one way to do it (which is by no means the most efficent):

First, let us choose a smallish prime, say 1009.

Our message will start with a header that says 1009^2 (1018081) in binary, followed by 1018081 zeros.

Next, we'll send about 9.8x10^93 "pages" each 1009 pixels wide and 1009 pixels high. To make sure the aliens figure out the format, we'll put a black margin around its page.

Now that we have a common visual platform, things should be easy. There are countless examples in both real life ("Cosmic Call") and fiction ("Contact") of how we can explain numbers and arithmetics in less than a dozen pages.

After basic arithmetics, we will need to explain set theory. This - again - shouldn't be difficult since we can use pictures. By page 1000 (probably much sooner), we should be able to speak with them freely about anything in ZF. By page 1000000 (probably much much sooner) we should be able to speak with them on anything mathematical, with the same ease we speak with humans.

Now, 1000000 pages is just a little more than 10^12 bits. We still have - pretty much - the same googol bits we had at the beginning. So the original question now boils down to:

"What is the largest number you can name with a googol bits,using only the widely accepted symbols of professional mathematics"

And since no human brain can process a googol bits in their lifetime, this can be further reduced to:

"What is the largest well-defined number you can name"

Which would be the current champion on the wiki (currently "Little Bigeddon").

We could send that one in less than a few trillion bits (probably far less). And if you really insist on using the entire googol bits, you could use them replace the usual input of Little Bigeddon (12^^12) with a definition of a larger argument. But that's just a naive extension, so it isn't really interesting.