Thank you for your question. I cannot tell for certain from the pictures that there are no seeds present, but you may be right. Resolution is just short of what is needed to see that. 

If indeed there are no seeds present there are only a couple situations I can envision that would cause that.

One would be serious water or heat stress. Really serious.

The other would be that you planted a hybrid variety. Hybrid varieties produce male sterile plants, and if there were no open pollinated carrots nearby to pollinate your carrots, you would get no seed. Do you know the variety? Where are you located? If you are located in western Oregon, pollen from wild carrots is abundant and this scenario is unlikely.

Let me know if this helps, or you need more information.