Sorry, no online dictionaries contain the exact phrase smallest infinite cardinal number.

Reverse dictionary results:
1. aleph-null
 2. one
3. transfinite number
4. six
5. twenty
6. dozen
7. numbering
8. numbing
9. zero
10. ten
 11. two
12. finite
13. infinity
14. five
15. nine
16. nineteen
17. seventeen
18. twelve
19. discrete
20. eight
 21. fifty
22. four
23. ninety
24. thirty
25. three
26. myriad
27. billion
28. million
29. least
30. thirteen
 31. phoebe
32. trillion
33. unit
34. iii
35. triad
36. leash
37. quintet
38. trey
39. trio
40. score

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You can look up the words in the phrase individually using these links:   smallest ?   infinite ?   cardinal ?   number ?
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