TITLE: How to Prove a Programming Language is Turing Complete?
QUESTION [49 upvotes]: I had some thoughts about how to prove the turing completeness of a programming language. I came to the conclusion, that if you could write a program that is able to parse a turing machine program, both should be equivalent as you could execute every turing machine program with that parser written in that language. Therefore the language must be turing complete. Am I right? What are other ways of proving turing completeness?

REPLY [3 votes]: Parsing is not to the point, and here is why. The language of all syntactically correct programs in a given language is (or should be) recursive, which is lower on the hierarchy than the languages defined by Turing-complete programming systems, which is recursively enumerable. In fact, for a very simple model (like Turing's original formulation), the language of all syntactically correct "programs" could be as low in the hierarchy as finite-state (regular).
In other words, there is very little relationship between the difficulty of parsing programs (a static kind of task) and simulating/emulating the running of a program (a very dynamic task).
So, as the others here say, you need to show that the programming language/model you want to prove Turing-complete can carry out certain basic operations and can also combine operations to make longer, arbitrarily complex programs.
Then there's the issue of the unlimited storage of theoretical models -- like the two-way unbounded tape of a classical Turing Machine -- versus the practical limitations on storage of real-life programming languages and systems. But that's a subtle topic that is usually not addressed in answering questions like yours, as it bears on a different set of theoretical issues.