  In this paper, we aim to conceptually examine the relationship between
logical incompleteness and concrete incompleteness which both study the
incompleteness phenomenon. We argue for two main theses. Firstly, the current
research on concrete incompleteness reals both similarities and differences
between logical incompleteness and concrete incompleteness. Similarities
between them are not universal, and differences between them are essential.
Secondly, concrete incompleteness is a higher order phenomenon over logical
incompleteness. This verifies that Hilbert's concrete and intuitive proof
theory provides us essential new information from non-concrete and
non-intuitive ideal proofs. We examine similarities between logical
incompleteness and concrete incompleteness from two aspects: equivalences
between logical incompleteness and concrete incompleteness, and the ubiquity of
the incompleteness phenomenon in both logical incompleteness and concrete
incompleteness. We examine differences between logical incompleteness and
concrete incompleteness from five aspects: (1) the influence on Hilbert's
program; (2) properties of independent sentences; (3) the intensionality
problem; (4) the relationship with ordinal analysis; (5) the limit of
provability.
