Not all ciphers can be broken, even by infinitely powerful adversaries.
When used correctly, the One Time Pad (OTP) is information-theoretic secure, which means it can't be broken with cryptanalysis. However, part of being provably secure is that you need at least as much key material as you have plaintext to encrypt. Such a key needs to be shared between the two communicants, which basically means you have to give it to the other person through a perfectly secure protocol (e.g. by hand/trusted courier). So, actually it just allows you to have your trusted meeting in advance, rather than at the time of transmitting the secret information.
To illustrate this, consider what happens if one tries to brute force OTP:
Since you have allowed an attacker infinite computational resources, he can keep guessing keys and calculating the corresponding plaintext until every key has been tested. Supposing the message was $b$ bits long, this would leave him with $2^b$ possible keys, each of which would generate a unique plaintext, making $2^b$ plaintexts. What is important here is that this means they would have candidate plaintexts corresponding to every possible bit-string of length $b$. This means, even if you knew the message was "Meet me at the stadium at 2
3), you still wouldn't have any idea what the
? was, because the possible plaintexts would contain this string with every possible value of
Most cryptographic methods we use now are computationally secure. There are lots of different ways to show this, and I'll just sketch two of them below:
We might come up with a reduction to a problem conjectured to be hard (e.g. the Diffie-Hellman Problem or Discrete Log Problem). That is, we prove that "If you can break my cipher, you can solve Some-Hard-Problem".
Meaning our problem is at least a difficult to solve as Some-Hard-Problem. So, if the problem is indeed hard to solve, then so must be cracking our encryption.
Another method used is that you might be able to prove a lower bound on the effort required to break a cipher, in terms of some security parameter. Then, the security parameter can be set such that this lower bound is higher than the strength of any computer that could possibly be built. Notice, the infinitely strong computer described in the question could still break such a scheme, but we could fix the parameter so high that it would be impossible to make a computer that could break it.
Possibly of interest: wikipedia:provable security