You Can’t Have Both! by Leo

Puzzle explanation

All grid entries were either inconsistent (clashing) or incomplete (lacking their last letter). The letters used to replace the clashes in the grid spelt INCONSISTENT from top to bottom. Clues for the incomplete entries contained a word with its last letter missing; these letters spelt INCOMPLETENESS. The five non-word grid entries were ASTR(O), AMEBOI(D), PEDDLIN(G), AIRMARSHA(L), UMPIRAG(E), whose last letters formed GÖDEL (omitting the umlaut was accepted). Kurt Gödel’s theorems state that systems that are complete (can prove all things) cannot be consistent (containing no contradictions) and ones that are consistent cannot be complete.

Clue explanations

Conventions: * = anagram, < = reversal

No Missing letter Answer Explanation
Across
1   ANTERIOR AN + TERROR with I for R; 2 meanings
6 fung(I) ATOPY A(dmitting) T(repidation) O(f) P(hoenix) Y(es)
9 ratio(N) KARAT RAT(ion) after KA
11   POMADE ((Brylcreem)ED A MOP(top)) <
13   SHUTTERED UTTER in SHED
14   TOIL TAIL with O for A; ref. poor batsmen
16   CONVECTIONAL (CONICAL VENT + O) *
17   UBER (T)UBE R(econstruction)
19 parse(C) ASTRO- AS + TR + O(bject)
22 Hob(O) TRAMP T(ime)R + AMP
24   TRON TR + ON
25   PREVENTATIVE (EVENT PRIVATE) *
29   SLEW 2 meanings
31   READINESS (DNA SERIES) *
32 Antony(M) ADDENDA ADD + END + À
33   GETA (GEISHA − (IS + H)) around T
34 ski(P) GUESS GUESTS − (pis)T(e)
35   REGARDER RE + (RED + RAG) <
Down
1   SKAT (Princes)S KAT(e)
2   TRIUNES TRIUNES + THEY = (THREES + UNITY) *
3   RETREATED RET + RE + A + TED
4   INURES INJURES − J
5   OTTO (c)OTTO(n)
6 cel(L) AMEBOID (BAD MOVIE) * − V
7 Borat(E) TAROC ((B)ORAT(e) + C) *
8 fee(T) PEDDLING (PINGED D(oorbel)L) *
10 pure(E) ALOOF (FOOL + A) <
12   DECANT DEC + ANT
15 patter(N) AIR-MARSHAL (LIAR HARMS A) *
17 grim(E) UMPIRAGE (UP A GRIME) *
18   HORNED HO + R + NE + D
20   GRIFTED GRIF(f) + TED
21   ABIDES E after (A + BID) + S
23   ATTEND A + T + T + END
26   VEIL (sur)VEIL(lance)
27   CODE C(omputable) O(r) D(ecidable) E(xpressions)
28 arm(S) UNDER UNDÉ + R
30 startles(S) WEARY (s)WEARY
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