Why "Infinite" Lossless Compression Fails: A Clarifying Framework

The phrase infinite lossless compression expresses a natural intuition: if compression removes redundancy, it can be tempting to imagine repeatedly compressing a compressed representation and continuing to shrink. This paper is primarily a clarification and formalization of that intuition rather than a report of new information-theoretic results. We articulate several precise questions that sit behind the informal phrase, distinguishing (i) worst-case lossless coding, (ii) coding for restricted, structured families of strings, and (iii) coding under a source distribution. Within this framework, we restate standard counting and entropy barriers in a self-contained way, and we explain how apparently “recursive” schemes typically relocate information into dictionaries, side information, or modeling assumptions. We also note simple settings in which compression ratios can diverge on restricted families without conflicting with classical bounds. The results presented here are not new to information theory; the contribution is a unified and intuitive framing of why repeated abstraction fails to yield unbounded lossless compression.