Can a person with delusions completely acknowledge their delusions I wanted to ask a question for a story I'm writing;

Can a person with delusions completely acknowledge that they have delusions? For example, if someone claims that for a long time he believed certain crucial things about his life to be true (possible things, nothing bizarre; Like his parents being dead and then seeing them at home, him going to a certain school but the school doesn't exist and he actually went to an entirely different one etc.) and then was proven otherwise. 

He has no fixations to his previous believes, he's completely open to the possibility of him not being able to differentiate reality (He was open the second he suspected something to be awry). 

Would he still fall under the delusional category? If not would it at least be considered an unordinary case?
The short answer to your question Can a person with delusions completely acknowledge their delusions is yes. But it depends. 

There are quite a few psychological disorders that have symptoms of delusion or hallucination. 

Bentall et al. (2009) took a look at the structure of paranoid delusions. They concluded that emotion related and cognitive processes are involved in paranoid delusions of participants. 



I think that a more complex answer to your question is that in the way you have described, the delusions would likely have been attempted to be shattered by loved ones and other people that are involved in the person with delusions life. 

Hallucinations can be symptoms of many psychological disorders such as: 


Unipolar and Bipolar Affective Disorders (Black &amp; Nasrallah, 1989)
Mood Disorders (Signer, et al., 1989)
Post Traumatic Stress Disorder (ptsd) (Scott et al., 2007)


This article may be very helpful to your story, the authors discuss delusional thought among non clinical patients i.e., people with no discernible psychological disorder (Freeman, 2006). 



References: 

Bentall, R. P., Rowse, G., Shryane, N., Kinderman, P., Howard, R., Blackwood, N., ... &amp; Corcoran, R. (2009). The cognitive and affective structure of paranoid delusions: a transdiagnostic investigation of patients with schizophrenia spectrum disorders and depression. Archives of general psychiatry, 66(3), 236-247. doi:10.1001/archgenpsychiatry.2009.1

Black D. W. &amp; Nasrallah A. (1989). Hallucinations and Delusions in 1,715 Patients with Unipolar and Bipolar Affective Disorders. Psychopathology 22(1), 28-34. doi: 10.1159/000284576

Freeman, D. (2006). Delusions in the nonclinical population
 Current Psychiatry Reports 8(3), 191-204. doi: 10.1007/s11920-006-0023-1

Scott, J. G., Nurcombe, B., Sheridan, J. &amp; McFarland M. (2007) Hallucinations in adolescents with post-traumatic stress disorder and psychotic disorder, Australasian Psychiatry, 15(1), 44-48, DOI: 10.1080/10398560601083084

Signer, S. F., Cummings, J. L., &amp; Benson, D. F. (1989). Delusions and mood disorders in patients with chronic aphasia. The Journal of Neuropsychiatry and Clinical Neurosciences, 1(1), 40-45.
doi: 10.1176/jnp.1.1.40